Analysis of Multiple Long-Run Relations in Panel Data Models

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-0.8inAnalysis of Multiple Long-Run Relations in Panel Data Models



\title{\vspace*{-0.8in}{\Large Analysis of Multiple Long-Run Relations in
Panel Data Models}\thanks{{\footnotesize The views expressed in this paper
are those of the authors and do not necessarily reflect those of the Federal
Reserve Bank of Dallas or the Federal Reserve System. We gratefully
acknowledge our use of computational resources provided by the Big-Tex High
Performance Computing Group at the Federal Reserve Bank of Dallas.}}}
\author{{\normalsize Alexander Chudik} \\
{\normalsize Federal Reserve Bank of Dallas} \and {\normalsize M. Hashem
Pesaran} \\
{\normalsize Trinity College, Cambridge, UK and University of Southern
California, USA} \and {\normalsize Ron P. Smith} \\
{\normalsize Birkbeck, University of London, UK}}
\date{
\normalsize
\today }
\maketitle

\begin{abstract}
The literature on panel cointegration is extensive but does not cover data
sets where the cross section dimension, $n$, is larger than the time series
dimension $T$. This paper proposes a novel methodology that filters out the
short run dynamics using sub-sample time averages as deviations from their
full-sample counterpart, and estimates the number of long-run relations and
their coefficients using eigenvalues and eigenvectors of the pooled
covariance matrix of these sub-sample deviations. We refer to this procedure
as pooled minimum eigenvalue (PME). We show that PME estimator is consistent
and asymptotically normal as $n$ and $T$ $\rightarrow \infty $ jointly, such
that $T\approx n^{d}$, with $d>0$ for consistency and $d>1/2$ for asymptotic
normality. Extensive Monte Carlo studies show that the number of long-run
relations can be estimated with high precision, and the PME estimators have
good size and power properties. The utility of our approach is illustrated
by micro and macro applications using Compustat and Penn World Tables.
\end{abstract}

\noindent \textbf{Keywords}: Multiple long-run relations, Pooled Minimum
Eigenvalue (PME) estimator, eigenvalue thresholding, panel data,
cointegration, interactive time effects, financial ratios, Penn World Table

\noindent \textbf{JEL Classification}: C13, C23, C33, G30.

\thispagestyle{empty}\pagebreak

\pagenumbering{arabic}

\section{Introduction}

\onehalfspacing
\normalsize

This paper provides a new methodology for the analysis of multiple long-run
relations in panel data models where the cross section dimension, $n$, is
large relative to the time series dimension, $T$. While there is an
extensive literature that considers multiple long-run (cointegrating)
relations for time series models, for panel data models with large $n$
researchers have mainly focussed on a single long-run relation with known
long-run causal links. The panel literature that does consider multiple
long-run relations assumes $n$ is fixed as $T\rightarrow \infty ,$ or adopt
sequential asymptotics whereby $T\rightarrow \infty $ first followed by $
n\rightarrow \infty $, effectively requiring $T$ to be large relative to $n$
and do not cover many applications of interest in economics and finance that
involve many cross section units, such as firms and countries, observed over
relatively short time spans. One example is empirical corporate finance,
which investigates the stability of long-run relations, including financial
ratios, using accounting data, such as Compustat, where thousands of firms
are observed over relatively few time periods.
\citeN{ColesLi2023}
provide examples from a number of sub-fields of corporate finance,
including: \ propensity to pay dividends, leverage, \ investment policy, and
firm performance. Another example is cross country empirical growth studies
that use data sets such as the Penn World Tables that provide annual data on
a range of macro variables for as many as $n=183$ countries over different
time periods, with a maximum time span of $T=70$. For both panel data sets
one would expect multiple long-run relations between the variables, some of
which may be the mean reverting ratios discussed in the macroeconomic and
finance literatures.

This paper proposes an estimation and testing strategy that applies to panel
data models with $n$ possibly much larger than $T$. We consider an $m\times
1 $ vector $\mathbf{w}_{it}$ for units $i=1,2,...,n$ over the time periods $
t=1,2,...,T$, with an unknown number, $r_{0}\in \left\{ 0,1,...,m-1\right\} $
, of linear combinations that are stationary. We refer to such linear
combinations as long-run relations. If it is known that all elements of $
\mathbf{w}_{it}$ are $I\left( 1\right) $ then the stationary relations can
be viewed as cointegrating relations. The focus of our analysis is to
estimate $r_{0}$, the number of common long-run relations, and their
coefficients, when $r_{0}\geq 1$. We filter out the short run dynamics by
means of $q$ ($\geq 2$) non-overlapping sub-sample time averages, $\mathbf{
\bar{w}}_{i\ell },$ $\ell =1,2,...,q,$ as deviations from their full-sample
counterpart, $\mathbf{\bar{w}}_{i\circ }$, namely $\mathbf{\bar{w}}_{i\ell }-
\mathbf{\bar{w}}_{i\circ }$, and then construct a pooled sample covariance
matrix of these deviations which we denote by $\mathbf{Q}_{\bar{w}\bar{w}}$.
The number of long-run relations and their coefficients are estimated using
the eigenvalues and eigenvectors of $\mathbf{Q}_{\bar{w}\bar{w}}$. We refer
to this procedure as pooled minimum eigenvalue (PME), and note that it is
simple to implement, extends readily to unbalanced panels, and is shown to
be robust to stationary interactive time effects. It is semi-parametric
since it does not require modelling the short run dynamics and applies to
general linear process, thus allowing for moving average processes and is
not confined to vector autoregressions (VAR). Most importantly, the PME
approach does not require knowing long-run causal linkages that might exist
amongst the variables under consideration. To our knowledge, no other panel
estimation procedure exists for such a setting.

Denoting the first $r_{0}$ eigenvectors of $\mathbf{Q}_{\bar{w}\bar{w}}$ by $
\mathbf{\hat{{\Greekmath 010C}}}_{j0},$ for $j=1,2,...,r_{0}$, we then consider
structural estimation of the long-run relations assuming they are subject to
$r_{0}\times r_{0}$ exact identifying restrictions. Assuming $r_{0}$ is
known, we derive the asymptotic distribution of the exactly identified
long-run relations and propose consistent estimators for their covariance
matrices that does not require estimation of the dynamics of individual $
\mathbf{w}_{it}$ processes; thus allowing us to test restrictions on the
elements of the exactly identified long-run relations with relatively short $
T$. The identified long-run relations are shown to be consistent and
asymptotically normally distributed as $n$ and $T$ $\rightarrow \infty $
jointly such that $T\approx n^{d}$. For consistency only $d>0$ is required,
but for asymptotic normality a faster relative rate of $d>1/2$ is required.
Many panel time series estimation and inference procedures require $d>1$ ($
n/T\rightarrow 0$). Our requirement $d>1/2$ indicates that the procedure
should work well with large $n$ and moderate $T$ allowing one to estimate
the coefficients of multiple long-run relations in such cases, without
estimating short-run dynamics.

We propose to estimate $r_{0}$ by the number of eigenvalues of $\mathbf{Q}_{
\bar{w}\bar{w}}$ that fall below a given threshold $C_{T}=CT^{-{\Greekmath 010E} }$,
for some $C>0$ and ${\Greekmath 010E} >0$. One could use cross validation procedures to
set $C$ and ${\Greekmath 010E} $, but based on extensive Monte Carlo experiments we
have found that setting $C=1$ works well \textit{if} we base our selection
procedure on the eigenvalues of the correlation matrix, $\mathbf{R}_{_{\bar{w
}\bar{w}}}=\left[ diag\left( \mathbf{Q}_{\bar{w}\bar{w}}\right) \right]
^{-1/2}\mathbf{Q}_{\bar{w}\bar{w}}\left[ diag\left( \mathbf{Q}_{\bar{w}\bar{w
}}\right) \right] ^{-1/2}$. Such an estimator can be written conveniently as
$\tilde{r}=\sum_{j=1}^{m}\mathcal{I}\left( \tilde{{\Greekmath 0115}}_{j}<T^{-{\Greekmath 010E}
}\right) ,$ where $\tilde{{\Greekmath 0115}}_{j},$ $j=1,2,...,m$ are the eigenvalues
of $\mathbf{R}_{_{\bar{w}\bar{w}}}$, and $\mathcal{I}\left( \mathcal{A}
\right) =1$ if $\mathcal{A}$ is true and zero otherwise. In the Monte Carlo
experiments and the empirical applications we report results for ${\Greekmath 010E}
=(1/4,1/2)$, and find that overall setting ${\Greekmath 010E} =1/4$ works well.

Monte Carlo experiments show near-perfect performance of $\tilde{r}$ as an
estimator of $r_{0}$, when ${\Greekmath 010E} =1/4$, for all $\,n=50$, $500$, $1,000$, $
3,000$ and $T=20$, $50$, $100$ sample size combinations and across a large
number of VAR and VARMA data generating processes, with and without
interactive time effects, non-Gaussian errors, generalized autoregressive
conditional heteroskedasticity (GARCH), threshold autoregressions (TAR), and
for different patterns of long-run causal ordering. $\tilde{r}$ performs
almost equally well for the smallest sample sizes of $T=20$ and $n=50$, as
well as for the largest $T=100$ and $n=3,000$. As an alternative approach we
considered Johansen's trace tests applied to each cross section unit
separately, and then estimated $r_{0}$ by the simple average of these
individual estimates. This was done purely for comparison since to the best
of our knowledge there are no other methods that apply to panels in the
literature.\ We found that this average type estimator performed reasonably
well when $T$ was large, but still fell short as compared to the
thresholding estimator.

The finite sample performance of PME estimator of the coefficients of the
long-run relations\ is found to be satisfactory with inference based on PME
estimator with $q=2$ (sub-sample time averages) generally more accurate in
terms of empirical size of the tests, compared with $q=4$, in line with
intuition that suggest a larger choice of $q$ is likely to result in a
larger finite-sample bias. We also considered a simple VAR(1) design with $
m=2$, $r_{0}=1$ and one-way long-run causality to see how PME performs
compared to the many single equation estimators proposed in the literature
(and cited below). We found that in this simple case the PME\ estimator is
less efficient in terms of root mean square errors only when $T=100$.
However, PME with $q=2$ proved to be less biased and performed much better
in terms of size than the single equation approaches for all sample size
combinations.

To illustrate the utility of PME procedure we present one micro and one
macro application. The micro application considers a number of key financial
variables (in logs) and investigates if they are cointegrated, and whether
financial ratios can be regarded as stationary variables. To this end we
used accounting data for individual firms from CRSP/Compustat on their book
value (BV), market value (MV), short-term debt (SD), long-term debt (LD),
total assets (TA) and total debt outstanding (DO). The panels involving
these variables are unbalanced and cover the period $1950-2021$. We consider
firms with at least $20$ years of data, with $n$ varying between about $
1,000 $ and $2,500$. The variables are grouped into three sets, where we
have prior expectations about possible cointegration and identification. The
first set considered has just two variables: the logarithm of total debt
outstanding and logarithm of total assets: \{$DO_{it}$, $TA_{it}$\}. The
ratio of total debt outstanding to total assets is often used as a measure
of leverage, which suggests a single hypothesized long-run relation. The
other two variable sets are: the logarithms of short and long term debt and
total assets, \{$SD_{it}$, $LD_{it}$, $TA_{it}$\}; and the logarithms of
total debt outstanding, book value and market value, \{$DO_{it}$, $BV_{it}$,
$MV_{it}$\}. We expect two hypothesized long-run relations in these sets
with three variables. For each set of variables we provide estimates for the
full sample 1950-2021 as well as for a shorter sample that ends in 2010. The
estimates provide strong evidence of one long-run relation when we consider
two variables, and, with one exception, two long-run relations when we
consider panels with three variables. In the case of panels with $m=2$, we
illustrate that the PME estimates are invariant to normalization, which is
in contrast to the estimates obtained using panel regressions that depend on
which way the regression is run. For the relation between logarithms of debt
and total assets, we find the estimates of the long-run coefficients are
close to one in all cases, ranging from $1.113$ to $1.143$, and precisely
estimated. In the case of panels with $m=3$ we find $\tilde{r}=2$, and the
null hypothesis that long-run coefficients are equal to unity is not
rejected in about a quarter of the panel estimates. These results provide
partial support for use of logarithm of financial ratio in corporate
finance. In cases where the use of log ratio is not supported, one could use
the PME estimates of long-run relations in second stage regressions on
stationary variables that also include short run dynamics as well as other
stationary variables.

The macro application investigates long-run relations using unbalanced cross
country macroeconomic time series data from the Penn World Tables, featuring
up to $n=177$ countries over the years $1950-2019$. This dataset has a much
smaller cross-section dimension and a larger average time dimension compared
with the micro application. We focus on four key macro variables: per capita
real merchandise exports ($ex_{it}$) and imports ($im_{it}$), real labour
productivity per hour worked ($prod_{it}$), and real wages per hour worked ($
wage_{it}$). The choice of these variables was motivated by two widely
maintained hypotheses. Firstly, real wages and productivity should balance
for steady state growth to be feasible. Secondly export and imports should
balance for international solvency, though the constraint may not be binding
for reserve-currency countries such as the US. These hypotheses are largely
confirmed for emerging economies, and, with notable departures from unit
long-run elasticities, also for advanced economies. In addition, when we
consider all the four variables together we uncover cross country evidence
on the long-run relation between exports and productivity without making any
assumption about the direction of causality between these variables.

\textbf{Related literature:} We first discuss the literature for a single
time series process, which could be viewed as the $p\times 1$ ($p=m$ $n$)
stacked vector, $\mathbf{w}_{t}=(\mathbf{w}_{1t}^{\prime },\mathbf{w}
_{2t}^{\prime },...,\mathbf{w}_{nt}^{\prime })^{\prime }$. Our approach is
related to that of
\citeANP{PhillipsOuliaris1990} (\citeyearNP{PhillipsOuliaris1988}, \citeyearNP{PhillipsOuliaris1990})
in that they also start from general linear processes. They propose testing
the null of no cointegration using the smallest eigenvalues of the spectral
density of $\Delta \mathbf{w}_{t}$ evaluated at zero frequency. However, it
is difficult to obtain reasonably precise estimates of the spectral density,
particularly in the presence of high persistence in first differences.
Attempting to eliminate the effects of the short run dynamics by using time
averages of sub-samples of time series data is also widely used.
\citeN{MuellerWatson2018}
consider using sub-sample averages to estimate the long-run relation between
two variables $(y_{t}$ and $x_{t})$. Their estimated long-run coefficient
from regression of sub-sample averages of $y_{t}$\ on those of $x_{t}$ \ is
not the same as the reciprocal of the estimate that will be obtained from
the reverses regression. A panel version of their procedure can be
considered, but will be subject to the same limitations, namely it can
handle only one long-run relation and will require knowing the direction of
long-run causality. In not requiring any assumptions regarding the direction
of long-run causality our approach is comparable to the maximum likelihood
approach pioneered by
\citeANP{Johansen1988} (\citeyearNP{Johansen1988}, \citeyearNP{Johansen1991a})
that allows for multiple long-run relations without assuming any long-run
causal orderings of the variables, but assumes a $VAR(s)$ specification in $
\mathbf{w}_{t}$ where $p$ and $s$ are fixed (and quite small) relative to $T$
.
\citeANP{Onatski2018} (\citeyearNP{Onatski2018}, \citeyearNP{Onatski2019})
\textbf{\ }investigate the asymptotic properties of Johansen test when $
\mathbf{w}_{t}$ follows VAR(1) but allow $p,T\rightarrow \infty ,$ such that
$p/T\rightarrow c\in (0,1]$. They provide theoretical arguments why
Johansen's test of cointegration rank is likely to be severely over-sized
even if $p$ takes moderate values. Extensions to higher order VARs are
provided by \cite{Bykhovskaya2022}. This is a promising approach which is
yet to be fully developed for the analysis of multiple cointegrations across
many units, which is the primary focus of this paper. Since the ordering of
the variables in the VAR does not affect the Johansen's tests of the
cointegration rank, without further restrictions the use of high-dimensional
VARs in $\mathbf{w}_{t}$ does not distinguish between cointegration across
units as compared to cointegration between the variables specific to the
cross section units. Also, the condition $p/T=nm/T\rightarrow c\in (0,1]$ is
unlikely to be met when $m>1$ and $n$ is of the same order of magnitude as $
T $.

Turning to the panel cointegration literature, most studies consider $I(1)$
variables with a single cointegrating vector where the direction of long-run
causality is known. These estimators are typically generalizations of the
time series procedures such as the panel Fully Modified OLS of
\citeANP{Pedroni1996} (\citeyearNP{Pedroni1996}, \citeyearNP{Pedroni2001}, \citeyearNP{Pedroni2001ReStat})
, the Pooled Mean Group (PMG) estimator of
\citeN{PesaranShinSmith1999}
,or the panel Dynamic OLS of
\citeN{MarkSul2003}
. There are panel generalizations of Johansen's approach, such as
\citeN{GroenKleibergen2003}
, and
\citeN{LarssonLyhagen2007}
, which can be used to test for the number of cointegrating relations and
estimate their parameters. These are based on a vector error correction
model, VECM, which can deal with multiple cointegrating vectors, but require
$T$ to be large relative to $n.$ In their applications
\citeN{LarssonLyhagen2007}
have $m=3,$ $n=4.$
\citeN{Breitung2005}
proposes a systems estimator, but that requires that every cross section
unit cointegrate.
\citeN{ChudikPesaranSmith2023}
suggest system pooled mean group estimator for a single common long-run
relation coefficient ${\Greekmath 0112} $, that can handle any long-run causal ordering
and allow some units to fail to cointegrate, but again requires $T$ to be
large relative to $n$. \

A large number of other topics have been examined within the context of a
panel with a single cointegrating relation. These include: estimation with
I(1) latent factors:
\citeN{BaiKaoNg2009}
and
\citeN{KapetaniosPesaranYamagata2011}
; structural breaks:
\citeN{BanerjeeCarrion2024}
and
\citeN{DitzenKaraviasWesterlund2025}
; and non-linear effects:
\citeN{deJongWagner2025}
. Further details can be found in the surveys by
\citeN{Breitung2008}
and
\citeN{Choi2015a}
that also cover testing for cointegration using residuals (
\citeANP{Westerlund2005}, \citeyearNP{Westerlund2005}
), and second generation panel unit root tests allowing for cross section
dependence (
\citeANP{Pesaran2007}, \citeyearNP{Pesaran2007}
). As this brief overview indicates, none of the methods advanced in the
literature consider multiple long-run relations when $n>>T$.

\textbf{Outline of the paper}: The rest of the paper is set out as follows:
Section \ref{Prelim} sets out the panel data model and introduces the PME
estimator. Section \ref{Assum} introduces the assumptions and discusses the
identification conditions. Section \ref{LongRR} gives a formal description
of the PME estimator and some of its asymptotic properties. Section \ref
{distribution}\ considers identification and derives the asymptotic
distribution of the exactly identified PME estimator. Section \ref{r_sel}
shows how $r_{0}$ can be estimated by eigenvalue thresholding. Section \ref
{IntEffects} allows for interactive time effects. Section \ref{qCTchoices}
discusses the choice of the number of sub-sample averages, $q$, and how to
set the parameters of the thresholding estimator of $r_{0}$. Section \ref{MC}
provides Monte Carlo evidence on the small sample properties of the PME
estimators of $r_{0}$ and $\mathbf{{\Greekmath 010C} }_{j0}$, $j=1,2,...,r_{0}$. Section
\ref{EA} discusses the empirical applications, and Section \ref{CON}
provides some concluding remarks. The proofs of the propositions and
theorems are provided in an appendix, with related lemmas given in a
supplement. This supplement also includes sub-sections on extensions of PME
to panels with interactive time effects, on how to implement the proposed
estimator for unbalanced panels, and gives details of the data generating
processes used in the Monte Carlo experiments, plus additional information
on data sources and the construction of the variables used in the empirical
applications.

\textbf{Notations:} Matrices are denoted by bold upper case letters and
vectors are denoted by bold lower case letters. All vectors are column
vectors. $\left\Vert \mathbf{x}\right\Vert $ denotes the Euclidean norm of a
vector $\mathbf{x}$. $rank\left( \mathbf{A}\right) $ denotes the column rank
of $\mathbf{A}$. $vec\left( \mathbf{A}\right) $ denotes vectorization of $
\mathbf{A}$. $tr(\mathbf{A})$ denotes the trace of a square matrix $\mathbf{A
}$. Eigenvalues of $m\times m$ symmetric positive semi-definite real matrix $
\mathbf{A}$ sorted in ascending order are $0\leq {\Greekmath 0115} _{1}(\mathbf{A}
)\leq {\Greekmath 0115} _{2}(\mathbf{A})\leq ...\leq {\Greekmath 0115} _{m}(\mathbf{A})$. $
\lVert \mathbf{A}\rVert $ is the spectral norm of $\mathbf{A}$. Small and
large finite positive constants that do not depend on sample sizes $n$ and $
T $ are denoted by ${\Greekmath 010F} $ and $K$, respectively. These constants can
take different values at different instances in the paper. $T_{n}\approx
n^{d}$ if {there exist $n_{0}\geq 1$ and positive constants }${\Greekmath 010F} ${\
and }$K${, such that }$\inf_{n\geq n_{0}}\left( T_{n}/n^{d}\right) \geq
{\Greekmath 010F} ${\ and $\sup_{n\geq n_{0}}\left( T_{n}/n^{d}\right) \leq K$.} {For
simplicity of exposition we omit subscript }$n$ and write $T\approx n^{d}$.
Convergence in probability and distribution are denoted by $\rightarrow _{p}$
and $\rightarrow _{d}$, respectively. In this paper $o_{p}\left( 1\right) $
is short for sequence of random variables, random vectors or random matrices
that converge to zero in probability as $n\rightarrow \infty $ for all
values of $d>1/2$. $\mathbf{A}_{n}=O_{p}\left( 1\right) $ if sequence $
\mathbf{A}_{n}$ is bounded in probability. $a_{n}=O(b_{n})$\emph{\ }denotes
the deterministic sequence $\left\{ a_{n}\right\} $\ is at most of order $
b_{n}$\emph{. }Equivalence of asymptotic distributions is denoted by $
\overset{a}{\thicksim }$.

\section{Preliminaries\label{Prelim}}

Consider the following general linear model for $\mathbf{w}_{it}$
\begin{equation}
\mathbf{w}_{it}=\mathbf{a}_{i}+\mathbf{G}_{i}\mathbf{f}_{t}+\mathbf{C}_{i}
\mathbf{s}_{it}+\mathbf{v}_{it},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }t=1,2,...,T;i=1,2,...,n,
\label{Grep}
\end{equation}
where $\mathbf{w}_{it}$ is an $m\times 1$ vector of outcomes, $\mathbf{a}
_{i} $ is $m\times 1$ vector of fixed effects, $\mathbf{f}_{t}$ is a vector
of stationary latent factors with associated loading matrices, $\mathbf{G}
_{i}$. $\mathbf{s}_{it}$ is the partial sum process defined by
\begin{equation}
\mathbf{s}_{it}=\mathbf{u}_{i1}+\mathbf{u}_{i2}+...+\mathbf{u}_{it},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{
for }t\geq 1\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }\mathbf{s}_{it}=\mathbf{0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, for }t<1,
\label{sit}
\end{equation}
$\mathbf{u}_{it}$ is independently distributed over $i$ and $t$ with mean
zero and the $m\times m$ positive definite matrix, $\mathbf{\Sigma }_{i}$, $
\mathbf{C}_{i}$ is an $m\times m$ matrix of fixed coefficients, and $\mathbf{
v}_{it}=\mathbf{C}_{i}^{\ast }(L)\mathbf{u}_{it}$, where $\mathbf{C}
_{i}^{\ast }(L)=\sum_{\ell =0}^{\infty }\mathbf{C}_{i\ell }^{\ast }L^{\ell }$
. This model covers many specifications of interest such as vector
autoregressions, error correction models, as well as first-differenced
stationary models. It allows for interactive time effects which reduce to
time effects under the so-called parallel trends assumption, namely setting $
\mathbf{G}_{i}=\mathbf{G}$ for all $i$. In stacked form the model for all $n$
units can be written as $\mathbf{w}_{t}=\mathbf{a+Gf}_{t}+\mathbf{Cs}_{t}+
\mathbf{C}^{\ast }(L)\mathbf{u}_{t},$ where $\mathbf{w}_{t}=(\mathbf{w}
_{1t}^{\prime },\mathbf{w}_{2t}^{\prime },...,\mathbf{w}_{nt}^{\prime
})^{\prime },$ $\mathbf{a=}\left( \mathbf{a}_{1}^{\prime },\mathbf{a}
_{2}^{\prime },...,\mathbf{a}_{n}^{\prime }\right) ^{\prime }$, $\mathbf{G=(G
}_{1}^{\prime },\mathbf{G}_{2}^{\prime },...,\mathbf{G}_{n}^{\prime
})^{\prime }$, and $\mathbf{u}_{t}=(\mathbf{u}_{1t}^{\prime },\mathbf{u}
_{2t}^{\prime },...,\mathbf{u}_{nt}^{\prime })^{\prime }$. Under our
specification $\mathbf{C}$ and $\mathbf{C}_{i}^{\ast }(L)$ are assumed to be
block-diagonal matrices with $\mathbf{C}_{i}$ and $\mathbf{C}_{i}^{\ast }(L)$
as their $i^{th}$ block, respectively. Such restrictions seem inevitable
when $n$ is large relative to $T$, and seems plausible considering that we
allow for cross-sectional dependence through the common factors, $\mathbf{f}
_{t}$.

Assuming that $\mathbf{C}_{i}$ has rank $m-r_{0}>0$ for all $i$, we are
interested in estimating $r_{0}$, and the associated stationary linear
combinations defined by $\mathbf{{\Greekmath 010C} }_{j0}^{\prime }\mathbf{w}_{it},$ $
j=1,2,...,r_{0},$ where $\mathbf{B}_{0}=\left( \mathbf{{\Greekmath 010C} }_{10},\mathbf{
{\Greekmath 010C} }_{20},...,\mathbf{{\Greekmath 010C} }_{r_{0}0}\right) $ is the $m\times r_{0}$
matrix of long-run relations that are common across all $i$, and satisfies $
\mathbf{B}_{0}^{\prime }\mathbf{C}_{i}=0$. We also consider estimation of
long-run relations subject to the exactly identifying restrictions that are
motivated by the theory. The estimator we propose involves splitting the
data for each unit into $q\geq 2$ sub-samples; taking time averages of these
sub-samples and forming a pooled demeaned covariance matrix we label $
\mathbf{Q}_{\bar{w}\bar{w}}$. The eigenvalues of this matrix allow us to
estimate $r_{0},$ and the eigenvectors corresponding to the first $r_{0}$
eigenvalues provide estimates of $\mathbf{{\Greekmath 010C} }_{j0}$. But to simplify the
exposition and focus on the main contribution of the paper, initially we
abstract from the interactive time effects, but return to this complication
in Section \ref{IntEffects}, where we show that our analysis remains valid
so long as the latent factors are stationary.

It is possible to allow for non-linear features, such as GARCH and threshold
autoregressions, so long as the effects of shocks to $\mathbf{u}_{it}$ decay
exponentially fast. But to keep the theoretical analyses relatively simple,
we only consider the robustness of our estimation and testing strategies to
such non-linear effects using Monte Carlo experiments.

\section{Assumptions and identification conditions\label{Assum}}

We directly work with (\ref{Grep}) and make the following assumptions:

\begin{assumption}
\textbf{\ }\label{ASS1} The error terms, $\mathbf{u}_{it}$, are distributed
independently over $i=1,2,...,n$ and $t=1,2,....,T$ with $E(\mathbf{u}_{it})=
\mathbf{0,}$ and the covariance matrix $\,E(\mathbf{u}_{it}\mathbf{u}
_{it}^{\prime })=\mathbf{\Sigma }_{i}$, where $\mathbf{\Sigma }_{i}$ is a
positive definite matrix, $\inf_{i}\mathbf{{\Greekmath 0115} }_{1}\left( \mathbf{
\Sigma }_{i}\right) >{\Greekmath 010F} $, $\sup_{i}\mathbf{{\Greekmath 0115} }_{m}\left(
\mathbf{\Sigma }_{i}\right) <K$, and $\sup_{it}E\left\Vert \mathbf{u}
_{it}\right\Vert ^{4+{\Greekmath 010F} }<K$, for some ${\Greekmath 010F} >0$.
\end{assumption}

\begin{assumption}
\label{ASS2} The coefficient matrices $\mathbf{C}_{i}$ and $\mathbf{C}
_{i\ell }^{\ast }$, are non-stochastic constants such that $
\sup_{i}\left\Vert \mathbf{C}_{i}\right\Vert <K$ and $\sup_{i}\left\Vert
\mathbf{C}_{i\ell }^{\ast }\right\Vert <K{\Greekmath 011A} ^{\ell }$, where ${\Greekmath 011A} $ lies
in the range $0<{\Greekmath 011A} <1$. The $m\times m$ matrix $\mathbf{C}_{i}$ has rank $
m-r_{0}$, for $i=1,2,...,n.$
\end{assumption}

\begin{assumption}
\label{ASS3} (a) Let
\begin{equation}
\mathbf{\Psi }_{n}=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\mathbf{\Sigma }_{i}
\mathbf{C}_{i}^{\prime }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }\mathbf{\Psi }=lim_{n\rightarrow \infty
}\mathbf{\Psi }_{n}.  \label{epsin1}
\end{equation}
Then there exists $n_{0}$ such that for all $n>n_{0}$, $rank(\mathbf{\Psi }
_{n})=rank(\mathbf{\Psi })=m-r_{0}>0$. (b) The orthonormalized eigenvectors
associated with the $r_{0}$ zero eigenvalues of $\ \left( \frac{q-1}{6q}
\right) \mathbf{\Psi }_{n}$\textbf{\ }are denoted by $\mathbf{{\Greekmath 010C} }_{j0}$,
for $j=1,2,...,r_{0}$, and the orthonormalized eigenvectors associated with
the ordered non-zero eigenvalues of $\left( \frac{q-1}{6q}\right) \mathbf{
\Psi }_{n}$, namely ${\Greekmath 0115} _{r_{0}+1}\leq {\Greekmath 0115} _{r_{0}+2}\leq ...\leq
{\Greekmath 0115} _{m}$, by $\mathbf{{\Greekmath 010C} }_{j}$, for $r_{0}+1,r_{0}+2,...,m$.
Specifically
\begin{equation}
\left( \frac{q-1}{6q}\right) \mathbf{\Psi }_{n}\mathbf{{\Greekmath 010C} }_{j0}=0\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,
for }j=1,2,...,r_{0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}  \label{coin0}
\end{equation}
and
\begin{equation}
\left( \frac{q-1}{6q}\right) \mathbf{\Psi }_{n}\mathbf{{\Greekmath 010C} }_{j}={\Greekmath 0115}
_{j}\mathbf{{\Greekmath 010C} }_{j}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, for }j=r_{0}+1,r_{0}+2,...,m\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\label{noncoin}
\end{equation}
\end{assumption}

\begin{remark}
Under Assumptions \ref{ASS1}-\ref{ASS2} $\left\{ \mathbf{v}_{it}\right\} $
has absolute summable autocovariances, $\sum_{h=0}^{\infty
}\sup_{i}\left\Vert \mathbf{\Gamma }_{i}(h)\right\Vert <K$, where $\mathbf{
\Gamma }_{i}(h)=E\left( \mathbf{v}_{it}\mathbf{v}_{i,t-h}^{\prime }\right) $
is the autocovariance function of $\mathbf{v}_{it}$. See Lemma \ref{Lacf} in
the supplement, Section \ref{A2}.
\end{remark}

\begin{remark}
Under Assumptions \ref{ASS3}, $\mathbf{{\Greekmath 010C} }_{j0}^{\prime }\mathbf{\Psi }
_{n}\mathbf{{\Greekmath 010C} }_{j0}=0$, for $j=1,2,...,r_{0}$ and ${\Greekmath 0115} _{j}=\left(
\frac{q-1}{6q}\right) \mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\Psi }_{n}\mathbf{
{\Greekmath 010C} }_{j}>0$, for $j=r_{0}+1,r_{0}+2,...,m$. Nonzero eigenvalues\textbf{\ }
${\Greekmath 0115} _{j}$, for $j=r_{0}+1,r_{0}+2,...,m$, and the corresponding
eigenvectors $\mathbf{{\Greekmath 010C} }_{j}$, for $j=r_{0}+1,r_{0}+2,...,m$, depend on
$n$, but to simplify the notations we avoid using the subscript $n$.
\end{remark}

\begin{remark}
\label{RemarkPP}Under part (b) of Assumption \ref{ASS3} $\mathbf{\Psi }_{n}$
and $\mathbf{\Psi }$ can be written as $\mathbf{\Psi }_{n}=\mathbf{P}
_{n}^{\prime }\mathbf{P}_{n}$, and $\mathbf{\Psi =P}^{\prime }\mathbf{P,}$
where $\mathbf{P}_{n}$ and $\mathbf{P}$ are $(m-r_{0})\times m$ full rank
matrices, with $rank\left( \mathbf{P}_{n}\right) =rank\left( \mathbf{P}
\right) =m-r_{0}$, for all $n>n_{0}$.
\end{remark}

\begin{remark}
Condition $rank\left( \mathbf{C}_{i}\right) =m-r_{0}$ for all $i=1,2,...,n$
in Assumption \ref{ASS2} can be relaxed to allow for no stochastic trends
for some cross section units, so long as the rank condition $rank\left(
\mathbf{\Psi }_{n}\right) =rank\left( \mathbf{\Psi }\right) =m-r_{0}$ holds.
Specifically, suppose that $\mathbf{C}_{i}=\mathbf{0}$ for $i=1,2,...,n_{1}$
, but the cointegration rank condition holds for the remaining units. Then
\begin{equation*}
\mathbf{\Psi }_{n}=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\mathbf{\Sigma }_{i}
\mathbf{C}_{i}^{\prime }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }=(1-{\Greekmath 0119} _{n})\left( \frac{1}{n-n_{1}}
\sum_{i=n_{1}+1}^{n}\mathbf{C}_{i}\mathbf{\Sigma }_{i}\mathbf{C}_{i}^{\prime
}\right) ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }
\end{equation*}
where ${\Greekmath 0119} _{n}=n_{1}/n$ is the proportion of units without stochastic
trends. Then the rank requirement continues to hold if ${\Greekmath 0119} _{n}>0$, and $
\frac{1}{n-n_{1}}\sum_{i=n_{1}+1}^{n}\mathbf{C}_{i}\mathbf{\Sigma }_{i}
\mathbf{C}_{i}^{\prime }$ tends to a matrix having rank $m-r_{0}$. But for
clarity of exposition we maintain Assumption \ref{ASS2} without loss of
generality.
\end{remark}

\section{Estimation of long-run relations\label{LongRR}}

\subsection{Introducing sub-sample time averages}

We base our estimation procedure on non-overlapping sub-sample time averages
of $\mathbf{w}_{it}$. For the ease of exposition, suppose the panel data
under consideration is balanced, $T$ is divisible by $q$, and consider $q$ $
(\geq 2)$ non-overlapping time averages of equal length $T_{q}$ defined by
\begin{equation}
\mathbf{\bar{w}}_{i\ell }=\frac{1}{T_{q}}\sum_{t=(\ell -1)T_{q}+1}^{\ell
T_{q}}\mathbf{w}_{it},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }\ell =1,2,...,q\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}  \label{la}
\end{equation}
where $T_{q}=T/q$. To simplify the exposition we abstract from interactive
effects and apply the above time average operator to (\ref{Grep}) to obtain
\footnote{
Sections \ref{Aunb} and \ref{TSIE} of the supplement consider unbalanced
panels and models with interactive effects.}
\begin{equation}
\mathbf{\bar{w}}_{i\ell }=\mathbf{a}_{i}+\mathbf{C}_{i}\mathbf{\bar{s}}
_{i\ell }+\mathbf{\bar{v}}_{i\ell }\mathbf{,}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }\ell =1,2,...,q,
\label{wbaris}
\end{equation}
where $\mathbf{\bar{v}}_{i\ell }=T_{q}^{-1}\sum_{t=(\ell -1)T_{q}+1}^{\ell
T_{q}}\mathbf{v}_{it}$ and $\mathbf{\bar{s}}_{i\ell
}=T_{q}^{-1}\sum_{t=(\ell -1)T_{q}+1}^{\ell T_{q}}\mathbf{s}_{it}.$ We now
use standard de-meaning procedure and eliminate $\mathbf{a}_{i}$ from (\ref
{wbaris}) to obtain
\begin{equation}
\mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }=\mathbf{C}_{i}\left(
\mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) +\left( \mathbf{
\bar{v}}_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right) ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }\ell
=1,2,...,q,  \label{wbar}
\end{equation}
where $\mathbf{\bar{w}}_{i\circ }=q^{-1}\sum_{\ell =1}^{q}\mathbf{\bar{w}}
_{i\ell }$, and similarly $\mathbf{\bar{s}}_{i\circ }=q^{-1}\sum_{\ell
=1}^{q}\mathbf{\bar{s}}_{i\ell }$ and $\mathbf{\bar{v}}_{i\circ
}=q^{-1}\sum_{\ell =1}^{q}\mathbf{\bar{v}}_{i\ell }$. Consider now the $
m\times m$ pooled sample covariance matrix
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}=n^{-1}\sum_{i=1}^{n}\mathbf{Q}_{\bar{w}_{i}\bar{w
}_{i}}  \label{Qbar}
\end{equation}
where
\begin{equation}
\mathbf{Q}_{\bar{w}_{i}\bar{w}_{i}}=T^{-1}q^{-1}\sum_{\ell =1}^{q}\left(
\mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) \left( \mathbf{
\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) ^{\prime }.
\label{Qibar}
\end{equation}
The limiting value of $\mathbf{Q}_{\bar{w}\bar{w}}$, as $n,T\rightarrow
\infty $, plays a critical role in our approach to estimation of long-run
relations. Using (\ref{wbar}) in (\ref{Qibar}) we first note that
\begin{equation}
\mathbf{Q}_{\bar{w}_{i}\bar{w}_{i}}=\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}
\bar{s}_{i}}\mathbf{C}_{i}^{\prime }+\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}
\bar{v}_{i}}+\mathbf{Q}_{\bar{v}_{i}\bar{s}_{i}}^{\prime }\mathbf{C}
_{i}^{\prime }+\mathbf{Q}_{\bar{v}_{i}\bar{v}_{i}},  \label{Qwiwi}
\end{equation}
where $T\mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}=q^{-1}\sum_{\ell =1}^{q}\left(
\mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) \left( \mathbf{
\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) ^{\prime },$ $T\mathbf{Q}
_{\bar{s}_{i}\bar{v}_{i}}=q^{-1}\sum_{\ell =1}^{q}\left( \mathbf{\bar{s}}
_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) \left( \mathbf{\bar{v}}_{i\ell }-
\mathbf{\bar{v}}_{i\circ }\right) ^{\prime }=T\mathbf{Q}_{\bar{v}_{i}\bar{s}
_{i}}^{\prime },$ and $T\mathbf{Q}_{\bar{v}_{i}\bar{v}_{i}}=q^{-1}\sum_{\ell
=1}^{q}\left( \mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right)
\left( \mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right) ^{\prime
} $. Since $\left\{ \mathbf{v}_{it}\right\} $ is covariance stationary with
absolute summable autocovariances and $\left\{ \mathbf{s}_{it}\right\} $ is
a partial sum process it then follows that $\mathbf{\bar{v}}_{i\ell }-
\mathbf{\bar{v}}_{i\circ }=$ $O_{p}\left( T^{-1/2}\right) $, and $\mathbf{
\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }=O_{p}(T^{1/2})$. Moreover, as
established in Lemma \ref{LQB},
\begin{equation}
\sup_{i}E\left\Vert \mathbf{Q}_{\bar{v}_{i}\bar{v}_{i}}\right\Vert =O\left(
T^{-2}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }\sup_{i}E\left\Vert \mathbf{Q}_{\bar{s}_{i}\bar{v}
_{i}}\right\Vert =O\left( T^{-1}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}  \label{sup_vv_sv}
\end{equation}

\subsection{Pooled minimum eigenvalue (PME) estimator}

Our proposed estimation procedure is based on eigenvalues and eigenvectors
of $\mathbf{Q}_{\bar{w}\bar{w}}$, defined by (\ref{Qbar}). Averaging $
\mathbf{Q}_{\bar{w}_{i}\bar{w}_{i}}$ in (\ref{Qwiwi}) over all cross section
units now yields:
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\mathbf{Q}_{
\bar{s}_{i}\bar{s}_{i}}\mathbf{C}_{i}^{\prime }+n^{-1}\sum_{i=1}^{n}\mathbf{C
}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}+n^{-1}\sum_{i=1}^{n}\mathbf{Q}_{
\bar{v}_{i}\bar{s}_{i}}^{\prime }\mathbf{C}_{i}^{\prime
}+n^{-1}\sum_{i=1}^{n}\mathbf{Q}_{\bar{v}_{i}\bar{v}_{i}}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\label{qwbd}
\end{equation}
The pooled minimum eigenvalue (PME) estimator of $\mathbf{{\Greekmath 010C} }_{j0},$ $
j=1,2,...,r_{0}$, is given by the $j^{th}$ orthonormalized eigenvector of $
\mathbf{Q}_{\bar{w}\bar{w}}$ , $\mathbf{\hat{{\Greekmath 010C}}}_{j}$, associated with
its $r_{0}$ smallest eigenvalues, $\hat{{\Greekmath 0115}}_{1}\leq \hat{{\Greekmath 0115}}
_{2}\leq ...\leq \hat{{\Greekmath 0115}}_{r_{0}}$. Specifically, for $j=1,2,....,m$, $
\mathbf{Q}_{\bar{w}\bar{w}}\mathbf{\hat{{\Greekmath 010C}}}_{j}\mathbf{=}\hat{{\Greekmath 0115}}
_{j}\mathbf{\hat{{\Greekmath 010C}}}_{j}$\textbf{, }such that $\mathbf{\hat{{\Greekmath 010C}}}
_{j}^{\prime }\mathbf{\hat{{\Greekmath 010C}}}_{j}=1$, and $\hat{{\Greekmath 0115}}_{j}=\mathbf{
\hat{{\Greekmath 010C}}}_{j}^{^{\prime }}\mathbf{Q}_{\bar{w}\bar{w}}\mathbf{\hat{{\Greekmath 010C}}}
_{j}$. In matrix notations we have
\begin{equation}
\mathbf{\hat{B}}=\left( \mathbf{\hat{{\Greekmath 010C}}}_{1},\mathbf{\hat{{\Greekmath 010C}}}
_{2},...,\mathbf{\hat{{\Greekmath 010C}}}_{m}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\left\Vert \mathbf{\hat{B}}
\right\Vert =1,  \label{PME}
\end{equation}
and the PME estimator of $\mathbf{B}_{0}$ is given by $\mathbf{\hat{B}}
_{0}=\left( \mathbf{\hat{{\Greekmath 010C}}}_{1},\mathbf{\hat{{\Greekmath 010C}}}_{2},...,\mathbf{
\hat{{\Greekmath 010C}}}_{r_{0}}\right) .$

\subsection{Consistency of the PME estimator\label{consistency}}

Under Assumption \ref{ASS1},$\ \mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}$ has the
following exact moment (established in Lemma \ref{LEQss})
\begin{equation}
E\left( \mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}\right) =\frac{(q-1)}{6}\left(
\frac{1}{q}+\frac{1}{T^{2}}\right) \mathbf{\Sigma }_{i}.  \label{EQsisi}
\end{equation}
Hence $n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}E\left( \mathbf{Q}_{\bar{s}_{i}\bar{
s}_{i}}\right) \mathbf{C}_{i}^{\prime }=\frac{(q-1)}{6}\left( \frac{1}{q}+
\frac{1}{T^{2}}\right) \mathbf{\Psi }_{n}\mathbf{,}$ where $\mathbf{\Psi }
_{n}$ is defined by (\ref{epsin1}), and by Assumption \ref{ASS3} is assumed
to have rank $m-r_{0}>0$. Furthermore, using $\sup_{i}\left\Vert \mathbf{C}
_{i}\right\Vert <K$ and (\ref{sup_vv_sv}) then
\begin{eqnarray}
n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}
&=&O_{p}\left( T^{-1}\right) ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }n^{-1}\sum_{i=1}^{n}\mathbf{Q}_{\bar{v}
_{i}\bar{s}_{i}}^{\prime }\mathbf{C}_{i}^{\prime }=O_{p}\left( T^{-1}\right)
,  \label{supQsv} \\
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and \ }n^{-1}\sum_{i=1}^{n}\mathbf{Q}_{\bar{v}_{i}\bar{v}_{i}}
&=&O_{p}\left( n^{-1/2}T^{-2}\right) .  \notag
\end{eqnarray}
Using the above results in (\ref{qwbd}) we have
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}-\frac{(q-1)}{6q}\mathbf{\Psi }
_{n}=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\left[ \mathbf{Q}_{\bar{s}_{i}\bar{s}
_{i}}-E\left( \mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}\right) \right] \mathbf{C}
_{i}^{\prime }+O_{p}\left( T^{-1}\right) .  \label{gapQww}
\end{equation}
Further $n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\left[ \mathbf{Q}_{\bar{s}_{i}
\bar{s}_{i}}-E\left( \mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}\right) \right]
\mathbf{C}_{i}^{\prime }=q^{-1}\sum_{\ell =1}^{q}\mathbf{G}_{\ell }-\mathbf{G
}_{0}$, where
\begin{equation*}
\mathbf{G}_{\ell }=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\left\{ \left( \frac{
\mathbf{\bar{s}}_{i\ell }}{\sqrt{T}}\right) \left( \frac{\mathbf{\bar{s}}
_{i\ell }}{\sqrt{T}}\right) ^{\prime }-E\left[ \left( \frac{\mathbf{\bar{s}}
_{i\ell }}{\sqrt{T}}\right) \left( \frac{\mathbf{\bar{s}}_{i\ell }}{\sqrt{T}}
\right) ^{\prime }\right] \right\} \mathbf{C}_{i}^{\prime }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ , \ for }
\ell =1,2,...,q,
\end{equation*}
and
\begin{equation*}
\mathbf{G}_{0}=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\left\{ \left( \frac{
\mathbf{\bar{s}}_{i\circ }}{\sqrt{T}}\right) \left( \frac{\mathbf{\bar{s}}
_{i\circ }}{\sqrt{T}}\right) ^{\prime }-E\left[ \left( \frac{\mathbf{\bar{s}}
_{i\circ }}{\sqrt{T}}\right) \left( \frac{\mathbf{\bar{s}}_{i\circ }}{\sqrt{T
}}\right) ^{\prime }\right] \right\} \mathbf{C}_{i}^{\prime }.
\end{equation*}
Under Assumptions \ref{ASS1} and \ref{ASS2}, $\mathbf{\bar{s}}_{i\ell }$ and
$\mathbf{\bar{s}}_{i\circ }$ are cross-sectionally independent random
variables and $\sup_{i}\left\Vert \mathbf{C}_{i}\right\Vert <K$. Further, $
T_{q}^{-1/2}\mathbf{\bar{s}}_{i\ell }$ and $T^{-1/2}\mathbf{\bar{s}}_{i\circ
}$ are scaled partial sums of $u_{it}\,\ $and tend to bounded random
variables. See, for example, result (d) of Proposition 17.1 in
\citeN{Hamilton1994}
. Therefore, $\mathbf{G}_{\ell }$ and $\mathbf{G}_{0}$ both converge at the
rate of $n^{-1/2}$ to their means that are zero, by construction. Namely $
\mathbf{G}_{\ell }=O_{p}\left( n^{-1/2}\right) $ and $\mathbf{G}
_{0}=O_{p}\left( n^{-1/2}\right) $. Hence
\begin{equation}
n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\left[ \mathbf{Q}_{\bar{s}_{i}\bar{s}
_{i}}-E\left( \mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}\right) \right] \mathbf{C}
_{i}^{\prime }=O_{p}\left( n^{-1/2}\right) ,  \label{CQss}
\end{equation}
and using this result in (\ref{gapQww}) yields
\begin{equation*}
\mathbf{Q}_{\bar{w}\bar{w}}-\frac{(q-1)}{6q}\mathbf{\Psi }_{n}=O_{p}\left(
n^{-1/2}\right) +O_{p}\left( T^{-1}\right) .
\end{equation*}
Therefore, for a fixed $q\left( \geq 2\right) $, $\mathbf{Q}_{\bar{w}\bar{w}
}\rightarrow _{p}\frac{(q-1)}{6q}\mathbf{\Psi }$, as $n,T\rightarrow \infty $
jointly such that $T_{n}\approx n^{d}$ and $d>0$, where $\mathbf{\Psi }
=lim_{n\rightarrow \infty }\mathbf{\Psi }_{n}$. This result is formally
established in the following proposition.

\begin{proposition}
\label{Qwbar}Consider the panel data model for $\mathbf{w}_{it}$ given by (
\ref{Grep}) without the interactive time effects ($\mathbf{G}_{i}=0$), and
suppose Assumptions \ref{ASS1} to \ref{ASS3} hold. Consider the $m\times m$
pooled sample covariance matrix $\mathbf{Q}_{\bar{w}\bar{w}}$ defined by (
\ref{Qbar}). Then$\ $ for $q\geq 2$ and a fixed $m$ we have
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}=\frac{(q-1)}{6q}\mathbf{\Psi }_{n}+O_{p}\left(
n^{-1/2}\right) +O_{p}\left( T^{-1}\right) ,  \label{Qbar2}
\end{equation}
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}\mathbf{{\Greekmath 010C} }_{j0}=O_{p}\left( n^{-1/2}\right)
+O_{p}\left( T^{-1}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, for }j=1,2,...,r_{0},  \label{QbarBeta}
\end{equation}
and $\mathbf{Q}_{\bar{w}\bar{w}}\rightarrow _{p}\frac{(q-1)}{6q}\mathbf{\Psi
}$, as $n,T\rightarrow \infty $ jointly such that $T_{n}\approx n^{d}$ and $
d>0$, where $\mathbf{\Psi }$ and $\mathbf{\Psi }_{n}$ are defined in
Assumption \ref{ASS3}.
\end{proposition}

For a known $r_{0}$ the PME estimator of $\mathbf{B}_{0}$, is given by $
\mathbf{\hat{B}}_{0}=\left( \mathbf{\hat{{\Greekmath 010C}}}_{1},\mathbf{\hat{{\Greekmath 010C}}}
_{2},...,\mathbf{\hat{{\Greekmath 010C}}}_{r_{0}}\right) $, where $\mathbf{\hat{{\Greekmath 010C}}}
_{j}$ , $j=1,2,...,m$ are the orthonormalized eigenvectors of $\mathbf{Q}_{
\bar{w}\bar{w}}$, as set out below equation (\ref{qwbd}). Then
\begin{equation}
\mathbf{\hat{B}}_{0}^{\prime }\mathbf{Q}_{\bar{w}\bar{w}}\mathbf{\hat{B}}
_{0}=\frac{(q-1)}{6q}\mathbf{\hat{B}}_{0}^{\prime }\mathbf{\Psi \hat{B}}
_{0}+O_{p}\left( n^{-1/2}\right) +O_{p}\left( T^{-1}\right) .  \label{a1}
\end{equation}
and $\mathbf{\hat{B}}_{0}^{\prime }\mathbf{Q}_{\bar{w}\bar{w}}\mathbf{\hat{B}
}_{0}$ is asymptotically minimized when $\mathbf{\hat{B}}_{0}^{\prime }
\mathbf{\Psi \mathbf{\hat{B}}}_{0}=\mathbf{0}$, and this occurs if $\mathbf{
\hat{B}}_{0}$ lies in the space spanned by the $r_{0}$ eigenvectors of $
\mathbf{\Psi }$ that are associated with its $r_{0}$ zero eigenvalues,
namely if and only if $\mathbf{\hat{B}}=\mathbf{\mathring{B}}_{0}\mathbf{H}$
, for some $r_{0}\times r_{0}$ non-singular rotation matrix, $\mathbf{H}$.
Hence, we have $\mathbf{\hat{B}}_{r}\mathbf{H}\rightarrow _{p}\mathbf{
\mathring{B}}_{0}$ as $n,T\rightarrow \infty $ jointly such that $T\approx
n^{d}$ and $d>0$. A formal statement is provided in the following theorem
with proofs in the Appendix.

\begin{theorem}
\label{Tcons}Consider the panel data model for the $m\times 1$ vector $
\mathbf{w}_{it}$ given by (\ref{Grep}) without the interactive time effects (
$\mathbf{G}_{i}=0$), and suppose that Assumptions \ref{ASS1} to \ref{ASS3}
hold and the number of long-run relations, $r_{0}$, is known. Let $\mathbf{
\hat{B}}_{0}=\left( \mathbf{\hat{{\Greekmath 010C}}}_{1},\mathbf{\hat{{\Greekmath 010C}}}_{2},...,
\mathbf{\hat{{\Greekmath 010C}}}_{r_{0}}\right) $ be the $m\times r_{0}$ matrix formed
from the orthonormalized eigenvectors of $\mathbf{Q}_{\bar{w}\bar{w}}$,
defined by (\ref{Qbar}), associated with its $r_{0}$ smallest eigenvalues.
Then for a fixed $m$ and $q$ $(\geq 2),$ $\mathbf{\hat{B}}_{0}\mathbf{
H\rightarrow }_{p}\mathbf{\mathring{B}}_{0}$ as $n,T\rightarrow \infty $
jointly such that $T_{n}\approx n^{d}$ and $d>0$, for any $r_{0}\times r_{0}$
non-singular matrix, $\mathbf{H}$.
\end{theorem}

\section{Identification and asymptotic distribution of PME estimator\label
{distribution}}

We focus on the case of exact identification of the long-run relations,
noting that the estimation of $r_{0}$ is invariant on how exact
identification is achieved.

\subsection{Exact identifying conditions\label{eic}}

We assume there exist $r_{0}^{2}$ a \textit{priori given and theoretically
meaningful }exact identifying restrictions on $\mathbf{B}_{0}$ given by
\begin{equation}
\mathbf{R}\mathbf{\mathring{B}}_{0}=\mathbf{(R}_{1},\mathbf{R}_{2})\left(
\begin{array}{c}
\mathbf{\mathring{B}}_{0,1} \\
\mathbf{\mathring{B}}_{0,2}
\end{array}
\right) \mathbf{=A,}  \label{ExactRes}
\end{equation}
where $\mathbf{\mathring{B}}_{0}$ is the $m\times r_{0}$ matrix of
identified long-run relations, $\mathbf{R}_{1}$,$\mathbf{R}_{2}$ and $
\mathbf{A}$ are $r_{0}\times r_{0},$ $r_{0}\times (m-r_{0})$ and $
r_{0}\times r_{0}$ matrices of known fixed constants, with $rank\left(
\mathbf{A}\right) =rank\left( \mathbf{R}_{1}\right) =r_{0}<m$. \ Then it
follows that $\mathbf{H}=\left( \mathbf{RB}_{0}\right) ^{-1}\mathbf{A}$, and
the PME estimator of $\mathbf{\mathbf{\mathring{B}}_{0}}$ is given by
\begin{equation}
\widehat{\mathbf{\mathbf{\mathring{B}}}}_{0}\mathbf{=\hat{B}}_{0}\left(
\mathbf{R\hat{B}}_{0}\right) ^{-1}\mathbf{A,}  \label{BETAdot_hat}
\end{equation}
where $\mathbf{\hat{B}}_{0}=\left( \mathbf{\hat{{\Greekmath 010C}}}_{10},\mathbf{\hat{
{\Greekmath 010C}}}_{20},...,\mathbf{\hat{{\Greekmath 010C}}}_{r_{0},0}\right) $. The exact
identifying restrictions, (\ref{ExactRes}), will often take the form $
\mathbf{\mathring{B}}_{0}=\left( \mathbf{I}_{r_{0}},\mathbf{\mathring{B}}
_{0,2}^{\prime }\right) ^{\prime }$, where $\mathbf{\mathring{B}}_{0,1}=
\mathbf{I}_{r_{0}}$\ is an identity matrix of order $r_{0}$. Without loss of
generality, we consider this formulation and denote $\mathbf{\mathring{B}}
_{0,2}$ by $\mathbf{\Theta }$. Once $\mathbf{\Theta }$ is estimated it is
possible to estimate $\mathbf{\mathring{B}}_{0,1}$ under more general
restrictions as $\mathbf{\mathring{B}}_{0,1}=\mathbf{R}_{1}^{-1}\left(
\mathbf{A-R}_{2}\mathbf{\Theta }\right) .$

\begin{proposition}
\label{PropID} Consider the $r_{0}^{2}$ exact identifying restrictions given
by (\ref{ExactRes}), and suppose $m\times r$ matrix of long-run relations $
\mathbf{\mathring{B}}_{0}$ is normalized as $\mathbf{\mathring{B}}
_{0}=\left( \mathbf{I}_{r_{0}},\mathbf{\Theta }^{\prime }\right) ^{\prime }$
, and Assumption \ref{ASS3} holds. Partition $\mathbf{\Psi }$ conformably
with $\mathbf{\mathring{B}}_{0}$ as $\mathbf{\Psi =}\left(
\begin{array}{cc}
\mathbf{\Psi }_{11} & \mathbf{\Psi }_{21}^{\prime } \\
\mathbf{\Psi }_{21} & \mathbf{\Psi }_{22}
\end{array}
\right) $. Then $\mathbf{I}_{r_{0}}+\mathbf{\Psi }_{11}$ and $\mathbf{\Psi }
_{22}$ are respectively $r_{0}\times r_{0}$ and $(m-r_{0})\times (m-r_{0})$
positive definite matrices and $\mathbf{\Theta }$ is uniquely determined by $
\mathbf{\Theta =-\Psi }_{22}^{-1}\mathbf{\Psi }_{21},$ subject to the
restrictions $\mathbf{\Psi }_{11}=\mathbf{\Psi }_{21}^{\prime }\mathbf{\Psi }
_{22}^{-1}\mathbf{\Psi }_{21}$. A proof is provided in the Appendix.
\end{proposition}

\subsection{Asymptotic distribution\label{AsyDis}}

To derive the asymptotic distribution of $\widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}$, note from (\ref{QwwD}) that
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left( \widehat{\mathbf{\mathring{B}}_{0}
}-\mathbf{\mathring{B}}_{0}\right) =-\left( n^{-1/2}\sum_{i=1}^{n}T\mathbf{C}
_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}\right) \mathbf{\mathring{B}}_{0}-
\frac{\sqrt{n}}{T}\left( n^{-1}\sum_{i=1}^{n}T^{2}\mathbf{Q}_{\bar{v}_{i}
\bar{v}_{i}}\right) \mathbf{\mathring{B}}_{0}+O_{p}\left( T^{1}\right)
\mathbf{.}  \label{qwb}
\end{equation}
By Lemma \ref{L_sqvvs}, $n^{-1}\sum_{i=1}^{n}T^{2}\mathbf{Q}_{\bar{v}_{i}
\bar{v}_{i}}=O_{p}\left( n^{-1/2}\right) $ and since $\left\Vert \mathbf{
\mathring{B}}_{0}\right\Vert <K$, then
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left( \widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}\right) =-\left( n^{-1/2}\sum_{i=1}^{n}T
\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}\right) \mathbf{\mathring{B}
}_{0}+O_{p}\left( T^{-1}\right) .  \label{DQww1}
\end{equation}
Further, write the first term as
\begin{eqnarray*}
\left( n^{-1/2}\sum_{i=1}^{n}T\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}
_{i}}\right) \mathbf{\mathring{B}}_{0} &=&\left( n^{-1/2}\sum_{i=1}^{n}\left[
T\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}-\mathbf{C}_{i}E\left( T
\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}\right) \right] \right) \mathbf{\mathring{
B}}_{0} \\
&&+\frac{\sqrt{n}}{T}\left( n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}E\left( T^{2}
\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}\right) \right) \mathbf{\mathring{B}}_{0},
\end{eqnarray*}
and using $\sup_{i}\left\Vert E\left( \mathbf{Q}_{\bar{s}_{i}\bar{v}
_{i}}\right) \right\Vert =O\left( T^{-2}\right) $ (established in Lemma \ref
{LQB}))
\begin{equation}
n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}E\left( T^{2}\mathbf{Q}_{\bar{s}_{i}\bar{v}
_{i}}\right) \mathbf{\mathring{B}}_{0}=O(1).  \label{T2qsvb}
\end{equation}
Using the above results in (\ref{DQww1}) we have
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left( \widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}\right) =-n^{-1/2}\sum_{i=1}^{n}\mathbf{Z}
_{i}+O_{p}\left( \frac{\sqrt{n}}{T}\right) ,  \label{DQww2}
\end{equation}
where $\mathbf{Z}_{i}=\mathbf{C}_{i}\left[ T\mathbf{Q}_{\bar{s}_{i}\bar{v}
_{i}}-E\left( T\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}\right) \right] \mathbf{
\mathring{B}}_{0}$. Recalling that $T\mathbf{Q}_{\bar{s}_{i}\bar{v}
_{i}}=q^{-1}\sum_{\ell =1}^{q}\left( \mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s
}}_{i\circ }\right) \left( \mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}
_{i\circ }\right) ^{\prime }$, then $\mathbf{Z}_{i}$ can be written as
\begin{equation}
\mathbf{Z}_{i}=\mathbf{C}_{i}\left[ q^{-1}\sum_{\ell =1}^{q}\left( \mathbf{
\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) \left( \mathbf{\bar{v}}
_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right) ^{\prime }\right] \mathbf{
\mathring{B}}_{0}-\mathbf{C}_{i}\left\{ q^{-1}\sum_{\ell =1}^{q}E\left[
\left( \mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) \left(
\mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right) ^{\prime }\right]
\right\} \mathbf{\mathring{B}}_{0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}  \label{zi}
\end{equation}
Writing (\ref{DQww2}) in vec form
\begin{equation*}
\left( \mathbf{I}_{r}\mathbf{\otimes Q}_{\bar{w}\bar{w}}\right) \sqrt{n}T
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ $vec$}\left( \widehat{\mathbf{\mathring{B}}}_{0}-\mathbf{\mathring{B}}
_{0}\right) =n^{-1/2}\sum_{i=1}^{n}\left( \mathbf{\mathring{B}}_{0}^{\prime }
\mathbf{\otimes \mathbf{C}}_{i}\right) \left[ \mathbf{\bar{{\Greekmath 0118}}}
_{iq}-E\left( \mathbf{\bar{{\Greekmath 0118}}}_{iq}\right) \right] +O_{p}\left( \frac{
\sqrt{n}}{T}\right) ,
\end{equation*}
where $\mathbf{\bar{{\Greekmath 0118}}}_{iq}=q^{-1}\sum_{\ell =1}^{q}\mathbf{{\Greekmath 0118} }_{i\ell
} $, and $\mathbf{{\Greekmath 0118} }_{i\ell }=vec\left[ \left( \mathbf{\bar{s}}_{i\ell }-
\mathbf{\bar{s}}_{i\circ }\right) \left( \mathbf{\bar{v}}_{i\ell }-\mathbf{
\bar{v}}_{i\circ }\right) ^{\prime }\right] =\left( \mathbf{\bar{v}}_{i\ell
}-\mathbf{\bar{v}}_{i\circ }\right) \mathbf{\otimes }\left( \mathbf{\bar{s}}
_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) $. Under Assumption \ref{ASS1}, $
\mathbf{\bar{v}}_{i\ell }$, $\mathbf{\bar{s}}_{i\ell }$, $\mathbf{\bar{v}}
_{i\circ }$ and $\mathbf{\bar{s}}_{i\circ }$ are distributed independently
over $i$. Hence, $\mathbf{\bar{{\Greekmath 0118}}}_{iq}$ is distributed independently over
$i$. In addition, $\sup_{i}\left\Vert \left( \mathbf{\mathring{B}}
_{0}^{\prime }\mathbf{\otimes \mathbf{C}}_{i}\right) \right\Vert =\left\Vert
\mathbf{\mathring{B}}_{0}\right\Vert \sup_{i}\left\Vert \mathbf{C}
_{i}\right\Vert <K$ and using Lemma \ref{Ld}, it follows that for $
(n,T)\rightarrow \infty \,,\ $jointly such that $T\approx n^{d}$ and $d>0$,
we have
\begin{equation}
n^{-1/2}\sum_{i=1}^{n}\left( \mathbf{\mathring{B}}_{0}^{\prime }\mathbf{
\otimes \mathbf{C}}_{i}\right) \left[ \mathbf{\bar{{\Greekmath 0118}}}_{iq}-E\left(
\mathbf{\bar{{\Greekmath 0118}}}_{iq}\right) \right] \rightarrow _{d}N\left( \mathbf{
0,\Omega }_{q}\right) ,  \label{egzd}
\end{equation}
where
\begin{equation}
\mathbf{\Omega }_{q}=lim_{n,T\rightarrow \infty }\left[ n^{-1}\sum_{i=1}^{n}
\left( \mathbf{\mathring{B}}_{0}^{\prime }\mathbf{\otimes \mathbf{C}}
_{i}\right) \mathbf{\Omega }_{\bar{{\Greekmath 0118}}_{iq}}\left( \mathbf{\mathring{B}}_{0}
\mathbf{\otimes \mathbf{C}}_{i}^{\prime }\right) \right] ,  \label{oz}
\end{equation}
and
\begin{equation}
\mathbf{\Omega }_{\bar{{\Greekmath 0118}}_{iq}}=Var\left( \mathbf{\bar{{\Greekmath 0118}}}_{iq}\right)
=q^{-2}\sum_{\ell =1}^{q}\sum_{\ell ^{\prime }=1}^{q}Cov\left( \mathbf{{\Greekmath 0118} }
_{i\ell },\mathbf{{\Greekmath 0118} }_{i\ell ^{\prime }}\right) .  \label{Omegaegzibar}
\end{equation}

Using (\ref{egzd}) in (\ref{DQww2}) yields asymptotic normality of $\widehat{
\mathbf{\mathring{B}}}$, which is formally established in the following
theorem.

\begin{theorem}
\label{Tb}Consider the panel data model for the $m\times 1$ vector $\mathbf{w
}_{it},$ given by (\ref{Grep}) without interactive time effects ($\mathbf{G}
_{i}=\mathbf{0}$). Suppose that Assumptions \ref{ASS1} to \ref{ASS3} hold, $
m $ and $q$ $(\geq 2)$ are fixed integers, and the number of long-run
relations, $r_{0}$ ($m>r_{0}>0$) is known. Suppose further that the long-run
relations, $\mathbf{\mathring{B}}_{0}$, of interest are subject to the exact
identifying restrictions, $\mathbf{R}\mathbf{\mathring{B}}_{0}\mathbf{=A}$
\textbf{, }given by (\ref{ExactRes}), and consider the PME estimator of $
\mathbf{\mathring{B}}_{0}$\textbf{\ }given by
\begin{equation*}
\widehat{\mathbf{\mathbf{\mathring{B}}}}_{0}\mathbf{=\hat{B}}_{0}\left(
\mathbf{R\hat{B}}_{0}\right) ^{-1}\mathbf{A,}
\end{equation*}
where $\mathbf{\hat{B}}_{0}=\left( \mathbf{\hat{{\Greekmath 010C}}}_{10},\mathbf{\hat{
{\Greekmath 010C}}}_{20},...,\mathbf{\hat{{\Greekmath 010C}}}_{r_{0},0}\right) $ are the first $
r_{0} $ orthonormalized eigenvectors of $\mathbf{Q}_{\bar{w}\bar{w}}$
defined by (\ref{Qbar}). Then
\begin{equation}
\sqrt{n}T\left( \mathbf{I}_{r_{0}}\mathbf{\otimes Q}_{\bar{w}\bar{w}}\right)
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{$vec$}\left( \widehat{\mathbf{\mathring{B}}}_{0}-\mathbf{\mathring{B}}
_{0}\right) \rightarrow _{d}N\left( \mathbf{0,\Omega }_{q}\right) ,
\label{dist}
\end{equation}
as $\left( n,T\right) \rightarrow \infty $, jointly such that $T\approx
n^{d} $ for $d>1/2$, where $\mathbf{\Omega }_{q}$ is defined by (\ref{oz}),
and $\mathbf{Q}_{\bar{w}\bar{w}}\rightarrow _{p}\frac{(q-1)}{6q}\mathbf{\Psi
}$. (see (\ref{Qbar2})). A proof is provided in the Appendix.
\end{theorem}

Theorem \ref{Tb} can be readily used to obtain asymptotic distribution of
any linear combination of $\widehat{\mathbf{\mathring{B}}}_{0}$. One notable
case of interest is to consider the exact identifying restrictions $\mathbf{
\mathring{B}}_{0,1}=\mathbf{I}_{r_{0}}$ discussed above. Under these
restrictions $\mathbf{\mathring{B}}_{0,1}=\widehat{\mathbf{\mathring{B}}}
_{0,1}=\mathbf{I}_{r_{0}}$, and $\widehat{\mathbf{\mathring{B}}}_{0}-\mathbf{
\mathring{B}}_{0}=\left(
\begin{array}{cc}
\mathbf{0} & \mathbf{\hat{\Theta}}^{\prime }-\mathbf{\Theta }_{0}^{\prime }
\end{array}
\right) $. Partitioning $\mathbf{Q}_{\bar{w}\bar{w}}$ and $\mathbf{\Omega }
_{z}$ accordingly, we have
\begin{equation*}
\sqrt{n}T\mathbf{Q}_{\bar{w}\bar{w}}\left( \widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}\right) =\left(
\begin{array}{cc}
\mathbf{Q}_{11,\bar{w}\bar{w}} & \mathbf{Q}_{12,\bar{w}\bar{w}} \\
\mathbf{Q}_{21,\bar{w}\bar{w}} & \mathbf{Q}_{22,\bar{w}\bar{w}}
\end{array}
\right) \left(
\begin{array}{c}
\mathbf{0} \\
\sqrt{n}T\left( \mathbf{\hat{\Theta}}-\mathbf{\Theta }_{0}\right)
\end{array}
\right) ,
\end{equation*}
where $\mathbf{Q}_{22,\bar{w}\bar{w}}\rightarrow _{p}\frac{(q-1)}{6q}\mathbf{
\Psi }_{22}$, and $\mathbf{\Psi }_{22}$ is the $(m-r_{0})\times (m-r_{0})$
lower right block of $\mathbf{\Psi }$ which is positive definite (see
Proposition \ref{PropID}). We obtain the following corollary.

\begin{corollary}
Suppose assumptions of Theorem \ref{Tb} hold and consider the exact
identifying restrictions $\mathbf{\mathring{B}}_{0,1}=\widehat{\mathbf{
\mathring{B}}}_{0,1}=\mathbf{I}_{r_{0}}$. Suppose $r_{0}$ is known, $q\geq 2$
, and let $\mathbf{\hat{\Theta}}$ and $\mathbf{\Theta }_{0}$ be the lower $
\left( m-r_{0}\right) \times r_{0}$ block of $\widehat{\mathbf{\mathring{B}}}
_{0}$ and $\mathbf{\mathring{B}}_{0}$, respectively. Then,
\begin{equation}
\sqrt{n}T\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ $vec$}\left( \mathbf{\hat{\Theta}}-\mathbf{\Theta }
_{0}\right) \rightarrow _{d}N\left( \mathbf{0,\Omega }_{{\Greekmath 0112} q}\right) ,
\label{Distheta}
\end{equation}
as $n,T\rightarrow \infty $, jointly such that $T\approx n^{d}$ for $d>1/2$
,where
\begin{equation}
\mathbf{\Omega }_{{\Greekmath 0112} q}=\left( \frac{6q}{q-1}\right) ^{2}\left( \mathbf{I
}_{r}\mathbf{\otimes \Psi }_{22}^{-1}\right) \mathbf{\Omega }_{q,22}\left(
\mathbf{I}_{r}\mathbf{\otimes \Psi }_{22}^{-1}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\label{Omegatheta}
\end{equation}
$\mathbf{\Omega }_{q,22}$ is the $\left( m-r_{0}\right) \times \left(
m-r_{0}\right) $ lower right block of $\mathbf{\Omega }_{q}$ given by (\ref
{oz}), and $\mathbf{\Psi }_{22}$ is the $(m-r_{0})\times (m-r_{0})$ lower
right block of $\mathbf{\Psi }$.
\end{corollary}

\subsection{Estimation of the asymptotic covariance of $\mathbf{\hat{\Theta}}
$}

To consistently estimate $\mathbf{\Omega }_{{\Greekmath 0112} q}$ we need a consistent
estimator of $\mathbf{\Omega }_{q,22}$, since $\left[ (q-1)/6q\right]
\mathbf{\Psi }_{22}$ can be consistently estimated by $\mathbf{Q}_{\bar{w}
\bar{w},22}$. Consider $\mathbf{\Omega }_{q}$ given by (\ref{oz}). Using (
\ref{wbar}) note that
\begin{eqnarray*}
\left( \mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) \left(
\mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) ^{\prime }
\mathbf{{\Greekmath 010C} }_{0} &=&\left[ \mathbf{C}_{i}\left( \mathbf{\bar{s}}_{i\ell }-
\mathbf{\bar{s}}_{i\circ }\right) +\left( \mathbf{\bar{v}}_{i\ell }-\mathbf{
\bar{v}}_{i\circ }\right) \right] \left[ \left( \mathbf{\bar{s}}_{i\ell }-
\mathbf{\bar{s}}_{i\circ }\right) ^{\prime }\mathbf{C}_{i}^{\prime }+\left(
\mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right) ^{\prime }\right]
\mathbf{\mathring{B}}_{0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,} \\
&=&\mathbf{C}_{i}\left( \mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ
}\right) \left( \mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right)
^{\prime }\mathbf{\mathring{B}}_{0}+\left( \mathbf{\bar{v}}_{i\ell }-\mathbf{
\bar{v}}_{i\circ }\right) \left( \mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}
_{i\circ }\right) ^{\prime }\mathbf{\mathring{B}}_{0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{eqnarray*}
Let $\mathbf{{\Greekmath 0110} }_{i\ell }=vec\left[ \left( \mathbf{\bar{w}}_{i\ell }-
\mathbf{\bar{w}}_{i\circ }\right) \left( \mathbf{\bar{w}}_{i\ell }-\mathbf{
\bar{w}}_{i\circ }\right) ^{\prime }\mathbf{\mathring{B}}_{0}\right] $, and $
\mathbf{{\Greekmath 0111} }_{i\ell }=vec\left[ \left( \mathbf{\bar{v}}_{i\ell }-\mathbf{
\bar{v}}_{i\circ }\right) \left( \mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}
_{i\circ }\right) ^{\prime }\right] $, where recall that $\mathbf{{\Greekmath 0111} }
_{i\ell }=O_{p}(T^{-1})$ uniformly in $i$ and $\ell $. Then $\mathbf{{\Greekmath 0110} }
_{i\ell }=\left( \mathbf{\mathring{B}}_{0}^{\prime }\mathbf{\otimes \mathbf{C
}}_{i}\right) \mathbf{{\Greekmath 0118} }_{i\ell }+\left( \mathbf{\mathring{B}}
_{0}^{\prime }\mathbf{\otimes I}_{m}\right) \mathbf{{\Greekmath 0111} }_{i\ell }$, and
\begin{equation*}
\mathbf{\bar{{\Greekmath 0110}}}_{iq}=q^{-1}\sum_{\ell =1}^{q}\mathbf{{\Greekmath 0110} }_{i\ell
}=\left( \mathbf{\mathring{B}}_{0}^{\prime }\mathbf{\otimes \mathbf{C}}
_{i}\right) \left( q^{-1}\sum_{\ell =1}^{q}\mathbf{{\Greekmath 0118} }_{i\ell }\right)
+\left( \mathbf{\mathring{B}}_{0}^{\prime }\mathbf{\otimes I}_{m}\right)
\left( q^{-1}\sum_{\ell =1}^{q}\mathbf{{\Greekmath 0111} }_{i\ell }\right) =\left(
\mathbf{\mathring{B}}_{0}^{\prime }\mathbf{\otimes \mathbf{C}}_{i}\right)
\mathbf{\bar{{\Greekmath 0118}}}_{iq}+O_{p}(T^{-1}),
\end{equation*}
where $\mathbf{\bar{{\Greekmath 0118}}}_{iq}=q^{-1}\sum_{\ell =1}^{q}\mathbf{{\Greekmath 0118} }_{i\ell
} $, as before. It follows
\begin{equation*}
n^{-1}\sum_{i=1}^{n}\mathbf{\bar{{\Greekmath 0110}}}_{iq}\mathbf{\bar{{\Greekmath 0110}}}
_{iq}^{\prime }=n^{-1}\sum_{i=1}^{n}\left( \mathbf{\mathring{B}}_{0}^{\prime
}\mathbf{\otimes \mathbf{C}}_{i}\right) \mathbf{\bar{{\Greekmath 0118}}}_{iq}\mathbf{\bar{
{\Greekmath 0118}}}_{iq}^{\prime }\left( \mathbf{\mathring{B}}_{0}\mathbf{\otimes \mathbf{C
}}_{i}^{\prime }\right) +O_{p}(T^{-1}),
\end{equation*}
and $E\left( \mathbf{\bar{{\Greekmath 0118}}}_{iq}\right) =O(T^{-1})$. Hence, as $
n,T\rightarrow \infty $ jointly such that $T\approx n^{d}$ for $d>1/2$,
\begin{equation*}
n^{-1}\sum_{i=1}^{n}\mathbf{\bar{{\Greekmath 0110}}}_{iq}\mathbf{\bar{{\Greekmath 0110}}}
_{iq}^{\prime }\rightarrow _{p}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }lim_{n,T}\left[ n^{-1}\sum_{i=1}^{n}
\left( \mathbf{\mathring{B}}_{0}^{\prime }\mathbf{\otimes \mathbf{C}}
_{i}\right) E\left( \mathbf{\bar{{\Greekmath 0118}}}_{iq}\mathbf{\bar{{\Greekmath 0118}}}_{iq}^{\prime
}\right) \left( \mathbf{\mathring{B}}_{0}\mathbf{\otimes \mathbf{C}}
_{i}^{\prime }\right) \right] =\mathbf{\Omega }_{q}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
Therefore, $\mathbf{\Omega }_{q}$ can be consistently estimated by the $
m^{2}\times m^{2}$ matrix
\begin{equation*}
\mathbf{\hat{\Omega}}_{q}=n^{-1}\sum_{i=1}^{n}\widehat{\mathbf{\bar{{\Greekmath 0110}}}}
_{iq}\widehat{\mathbf{\bar{{\Greekmath 0110}}}}_{iq}^{\prime }=\left(
\begin{array}{cc}
\mathbf{\hat{\Omega}}_{q,11} & \mathbf{\hat{\Omega}}_{q,12} \\
\mathbf{\hat{\Omega}}_{q,21} & \mathbf{\hat{\Omega}}_{q,22}
\end{array}
\right) ,
\end{equation*}
where
\begin{equation*}
\widehat{\mathbf{\bar{{\Greekmath 0110}}}}_{iq}=q^{-1}\sum_{\ell =1}^{q}vec\left[ \left(
\mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) \left( \mathbf{
\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) ^{\prime }\widehat{
\mathbf{\mathring{B}}_{0}}\right] =q^{-1}\sum_{\ell =1}^{q}\left[ \widehat{
\mathbf{\mathring{B}}}_{0}^{\prime }\left( \mathbf{\bar{w}}_{i\ell }-\mathbf{
\bar{w}}_{i\circ }\right) \mathbf{\otimes I}_{m}\right] \left( \mathbf{\bar{w
}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
Let $\widehat{\mathbf{\mathring{B}}_{0}}^{\prime }\left( \mathbf{\bar{w}}
_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) =\widehat{\mathbf{\bar{E}}}
_{i\ell }$, the $r\times 1$ vector of error corrections for the sub-sample $
\ell $, then
\begin{equation}
\mathbf{\hat{\Omega}}_{q}=n^{-1}\sum_{i=1}^{n}q^{-2}\sum_{\ell
=1}^{q}\sum_{\ell ^{\prime }=1}^{q}\left( \widehat{\mathbf{\bar{E}}}_{i\ell }
\mathbf{\otimes I}_{m}\right) \left( \mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w
}}_{i\circ }\right) \left( \mathbf{\bar{w}}_{i\ell ^{\prime }}-\mathbf{\bar{w
}}_{i\circ }\right) ^{\prime }\left( \widehat{\mathbf{\bar{E}}}_{i\ell
^{\prime }}^{\prime }\mathbf{\otimes I}_{m}\right) .  \label{Omegaz}
\end{equation}
Using the above results we now have
\begin{equation}
\widehat{Var\left[ vec\left( \mathbf{\hat{\Theta}}\right) \right] }=\frac{1}{
nT^{2}}\mathbf{Q}_{\bar{w}\bar{w},22}^{-1}\mathbf{\hat{\Omega}}_{q,22}
\mathbf{Q}_{\bar{w}\bar{w},22}^{-1}.  \label{VarHatTheta}
\end{equation}

When $r_{0}=1$, estimate of the error correction term $\widehat{\mathbf{
\mathring{B}}}_{0}^{\prime }\left( \mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}
_{i\circ }\right) =\widehat{\bar{e}}_{i\ell }$ is a scalar and we can write $
\widehat{\mathbf{\bar{{\Greekmath 0110}}}}_{iq}=q^{-1}\sum_{\ell =1}^{q}\left( \mathbf{
\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) \widehat{\bar{e}}_{i\ell
}$, and
\begin{eqnarray*}
\mathbf{\hat{\Omega}}_{q} &=&n^{-1}\sum_{i=1}^{n}\left[ q^{-1}\sum_{\ell
=1}^{q}\left( \mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right)
\widehat{\bar{e}}_{i\ell }\right] \left[ q^{-1}\sum_{\ell =1}^{q}\left(
\mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right) ^{\prime }
\widehat{\bar{e}}_{i\ell }\right] \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,} \\
&=&n^{-1}\sum_{i=1}^{n}\left[ q^{-2}\sum_{\ell =1}^{q}\sum_{\ell ^{\prime
}=1}^{q}\left( \mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right)
\left( \mathbf{\bar{w}}_{i\ell ^{\prime }}-\mathbf{\bar{w}}_{i\circ }\right)
^{\prime }\widehat{\bar{e}}_{i\ell }\widehat{\bar{e}}_{i\ell ^{\prime }}
\right] \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\end{eqnarray*}
which resembles the robust covariance matrix estimator that arises in
estimation of panel data models with short $T$ and large $n$. Here $q$ plays
the role of $T$.

\section{Estimation of $r_{0}$ by eigenvalue thresholding\label{r_sel}}

Under Assumption \ref{ASS3}, the true number of common long-run relations, $
r_{0}$, is defined by $rank(\mathbf{\Psi }_{n})=m-r_{0}>0$, where $\mathbf{
\Psi }_{n}=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\mathbf{\Sigma }_{i}\mathbf{C}
_{i}^{\prime }$. Subject to this condition, $\mathbf{\Psi }_{n}\mathbf{{\Greekmath 010C}
}_{j,0}=0$, for $j=1,2,...,r_{0}$, where $\mathbf{{\Greekmath 010C} }_{j,0}$ is the $
j^{th}$ long-run relation ($j\leq r_{0}$). The $m\times r_{0}$ matrix of
long-run relations is denoted by $\mathbf{B}_{0}$. It is also worth bearing
in mind that under Assumption \ref{ASS3}, $\mathbf{\Psi }_{n}\mathbf{{\Greekmath 010C} }
_{j}\neq \mathbf{0}$, for $j=r_{0}+1,r_{0}+2,...m$, namely cannot be spanned
by the $r_{0}$ columns of $\mathbf{B}_{0}$. See (\ref{coin0}) and (\ref
{noncoin}). There is a clear shift in the ordered eigenvalues of $\mathbf{
\Psi }_{n}$ from ${\Greekmath 0115} _{r_{0}}=$ $0$ to ${\Greekmath 0115} _{r_{0}+1}>0$, which
allows us to propose a thresholding estimator of $r_{0}$ applied to the
eigenvalues of $\mathbf{Q}_{\bar{w}\bar{w}}$, noting that under our
assumptions $\mathbf{Q}_{\bar{w}\bar{w}}$ tends to $\left( \frac{q-1}{6q}
\right) \mathbf{\Psi }$ as $n$ and $T\rightarrow \infty $. See result (\ref
{Qbar2}) of Proposition \ref{Qwbar}. Such an estimator can be written
conveniently as
\begin{equation}
\hat{r}=\sum_{j=1}^{m}\mathcal{I}\left( \hat{{\Greekmath 0115}}_{j}<C_{T}\right) ,
\label{rhat}
\end{equation}
where $\hat{{\Greekmath 0115}}_{1}\leq \hat{{\Greekmath 0115}}_{2}\leq ....\leq \hat{{\Greekmath 0115}}
_{m} $ are ordered eigenvalues of $\mathbf{Q}_{\bar{w}\bar{w}}$, and $
\mathcal{I}\left( \mathcal{A}\right) =1$ if $\mathcal{A}$ is true or zero
otherwise, and $C_{T}=KT^{-{\Greekmath 010E} }$, for some ${\Greekmath 010E} >0$. This estimator
is invariant to the ordering of the eigenvalues, but using the ordering
helps with the exposition and the rationale behind the proofs.

To establish the consistency of $\hat{r}$ as an estimator of $r_{0}$, we
first note that (\ref{rhat}) can be written equivalently as
\begin{equation}
\hat{r}-r_{0}=-\sum_{j=1}^{r_{0}}\mathcal{I}\left( \hat{{\Greekmath 0115}}_{j}\geq
C_{T}\right) +\sum_{j=r_{0}+1}^{m}\mathcal{I}\left( \hat{{\Greekmath 0115}}
_{j}<C_{T}\right) ,  \label{rhatgap}
\end{equation}
which in turn yields:
\begin{equation}
E\left\vert \hat{r}-r_{0}\right\vert \leq \sum_{j=1}^{r_{0}}\Pr \left( \hat{
{\Greekmath 0115}}_{j}\geq C_{T}\right) +\sum_{j=r_{0}+1}^{m}\Pr \left( \hat{{\Greekmath 0115}}
_{j}<C_{T}\right) .  \label{egap}
\end{equation}
Again noting the ordering of the eigenvalues, $\Pr \left( \hat{{\Greekmath 0115}}
_{j}\geq C_{T}\right) \leq \Pr \left( \hat{{\Greekmath 0115}}_{1}\geq C_{T}\right) $,
for $j=2,3,...,r_{0},$ and $\Pr \left( \hat{{\Greekmath 0115}}_{r_{0}+1}<C_{T}\right)
\geq \Pr \left( \hat{{\Greekmath 0115}}_{j}<C_{T}\right) $, for $
j=r_{0}+2,r_{0}+3,...,m$. Using these results in (\ref{egap}) we have
\begin{equation}
E\left\vert \hat{r}-r_{0}\right\vert \leq r_{0}\Pr \left( \hat{{\Greekmath 0115}}
_{1}\geq C_{T}\right) +(m-r_{0})\Pr \left( \hat{{\Greekmath 0115}}_{r_{0}+1}<C_{T}
\right) .  \label{Eup}
\end{equation}
Similarly,
\begin{eqnarray}
E\left( \hat{r}-r_{0}\right) ^{2} &\leq &r_{0}^{2}\Pr \left( \hat{{\Greekmath 0115}}
_{1}\geq C_{T}\right) +(m-r_{0})^{2}\Pr \left( \hat{{\Greekmath 0115}}
_{r_{0}+1}<C_{T}\right)  \label{MSErhat} \\
&&+2r_{0}(m-r_{0})\sqrt{\Pr \left( \hat{{\Greekmath 0115}}_{1}\geq C_{T}\right) \Pr
\left( \hat{{\Greekmath 0115}}_{r_{0}+1}<C_{T}\right) }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}  \notag
\end{eqnarray}
Also, by Markov inequality there exists ${\Greekmath 010F} >0$ such that
\begin{equation}
\Pr \left( \left\vert \hat{r}-r_{0}\right\vert >{\Greekmath 010F} \right) \leq
(r_{0}/{\Greekmath 010F} )\Pr \left( \hat{{\Greekmath 0115}}_{1}\geq C_{T}\right) +\left[
(m-r_{0})/{\Greekmath 010F} \right] \Pr \left( \hat{{\Greekmath 0115}}_{r_{0}+1}<C_{T}\right) ,
\label{Prrhat}
\end{equation}
Again by Markov inequality $\Pr \left( \hat{{\Greekmath 0115}}_{1}\geq C_{T}\right)
\leq C_{T}^{-1}E\left( \hat{{\Greekmath 0115}}_{1}\right) $, and using result (\ref{el}
) established in Lemma \ref{LQb}, we have
\begin{equation}
\Pr \left( \hat{{\Greekmath 0115}}_{1}\geq C_{T}\right) =O\left(
C_{T}^{-1}n^{-1/2}T^{-2}\right) .  \label{PrL1}
\end{equation}
Consider now $\Pr \left( \hat{{\Greekmath 0115}}_{r_{0}+1}<C_{T}\right) $, and recall
that
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}\mathbf{\hat{{\Greekmath 010C}}}_{j}=\hat{{\Greekmath 0115}}_{j}\mathbf{
\hat{{\Greekmath 010C}}}_{j}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }\hat{{\Greekmath 0115}}_{j}=\mathbf{\hat{{\Greekmath 010C}}}
_{j}^{\prime }\mathbf{Q}_{\bar{w}\bar{w}}\mathbf{\hat{{\Greekmath 010C}}}_{j}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, for
}j=r_{0}+1,r_{0}+2,...,m,  \label{ljhat}
\end{equation}
with associated population values given by (see Assumption \ref{ASS3}).
\begin{equation}
\frac{(q-1)}{6q}\mathbf{\Psi }_{n}\mathbf{{\Greekmath 010C} }_{j}={\Greekmath 0115} _{j}\mathbf{
{\Greekmath 010C} }_{j}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }\frac{(q-1)}{6q}\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{
\Psi }_{n}\mathbf{{\Greekmath 010C} }_{j}={\Greekmath 0115} _{j}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textbf{, }for }
j=r_{0}+1,r_{0}+2,...,m.  \label{lj}
\end{equation}
where $\mathbf{\Psi }_{n}=n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\mathbf{\Sigma }
_{i}\mathbf{C}_{i}^{\prime }\succeq 0$, and $\mathbf{{\Greekmath 010C} }_{j}^{\prime }
\mathbf{{\Greekmath 010C} }_{j}=1$. Furthermore, ${\Greekmath 0115} _{j}>0$ and $\mathbf{\Psi }_{n}
\mathbf{{\Greekmath 010C} }_{j}\neq 0,$ for $j>r_{0}$. But using (\ref{qwbd}) and
results established in Lemma \ref{LQB} and Proposition \ref{Qwbar} we have $
E\left( \mathbf{Q}_{\bar{w}\bar{w}}\right) =\frac{(q-1)}{6q}\mathbf{\Psi }
_{n}+O\left( T^{-2}\right) $, and
\begin{equation*}
\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}=\mathbf{Q}_{\bar{w}\bar{w}}-E\left(
\mathbf{Q}_{\bar{w}\bar{w}}\right) =n^{-1}\sum_{i=1}^{n}\mathbf{C}_{i}\left[
\mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}-E\left( \mathbf{Q}_{\bar{s}_{i}\bar{s}
_{i}}\right) \right] \mathbf{C}_{i}^{\prime }+O_{p}\left( T^{-1}\right)
=O_{p}\left( n^{-1/2}\right) +O_{p}\left( T^{-1}\right) .
\end{equation*}
Using the above results then (\ref{lj}) can be written equivalently as $
E\left( \mathbf{Q}_{\bar{w}\bar{w}}\right) \mathbf{{\Greekmath 010C} }_{j}={\Greekmath 0115} _{j}
\mathbf{{\Greekmath 010C} }_{j}+O\left( T^{-2}\right) $,$\,$\ and ${\Greekmath 0115} _{j}=\mathbf{
{\Greekmath 010C} }_{j}^{\prime }E\left( \mathbf{Q}_{\bar{w}\bar{w}}\right) \mathbf{
{\Greekmath 010C} }_{j}+O\left( T^{-2}\right) $. Therefore, together with (\ref{ljhat}),
we have
\begin{equation*}
\hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}=\mathbf{\hat{{\Greekmath 010C}}}_{j}^{\prime }\mathbf{Q}_{
\bar{w}\bar{w}}\mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}^{\prime }E\left(
\mathbf{Q}_{\bar{w}\bar{w}}\right) \mathbf{{\Greekmath 010C} }_{j}+O\left( T^{-2}\right)
\end{equation*}
and rearranged as
\begin{eqnarray}
&&\left( \left[ E\left( \mathbf{Q}_{\bar{w}\bar{w}}\right) -\mathbf{I}
_{m}{\Greekmath 0115} _{j}\right] \right) \left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{
{\Greekmath 010C} }_{j}\right) -\left( \hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) \mathbf{
{\Greekmath 010C} }_{j}  \label{betaja2} \\
&=&-\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}\mathbf{{\Greekmath 010C} }_{j}+\left( \hat{
{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) \left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{
{\Greekmath 010C} }_{j}\right) -\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}\left( \mathbf{\hat{
{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) +O\left( T^{-2}\right) .  \notag
\end{eqnarray}
Also, since $\mathbf{{\Greekmath 010C} }_{j}^{\prime }E\left( \mathbf{Q}_{\bar{w}\bar{w}
}\right) ={\Greekmath 0115} _{j}\mathbf{{\Greekmath 010C} }_{j}^{\prime }+O\left( T^{-2}\right) ,$
then \textbf{\ }
\begin{equation}
\hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}=\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\tilde{Q}
}_{\bar{w}\bar{w}}\mathbf{{\Greekmath 010C} }_{j}+2{\Greekmath 0115} _{j}\mathbf{{\Greekmath 010C} }
_{j}^{\prime }\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) +2
\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}\left(
\mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) +\left( \mathbf{\hat{
{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) ^{\prime }\mathbf{Q}_{\bar{w}\bar{w}
}\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) +O\left(
T^{-2}\right) .  \label{lambdaja2}
\end{equation}
where $\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}=O_{p}\left( n^{-1/2}\right)
+O_{p}\left( T^{-1}\right) $. Pre-multiplying both sides of the above
equations by $\sqrt{n}$ and stacking them in matrix notation we will have
\begin{equation*}
\left(
\begin{array}{cc}
1 & -2{\Greekmath 0115} _{j}\mathbf{{\Greekmath 010C} }_{j}^{\prime } \\
-\mathbf{{\Greekmath 010C} }_{j} & E\left( \mathbf{Q}_{\bar{w}\bar{w}}\right) -{\Greekmath 0115}
_{j}\mathbf{I}_{m}
\end{array}
\right) \left(
\begin{array}{c}
\sqrt{n}\left( \hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) \\
\sqrt{n}\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right)
\end{array}
\right) =\left(
\begin{array}{c}
\sqrt{n}\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}
\mathbf{{\Greekmath 010C} }_{j} \\
-\sqrt{n}\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}\mathbf{{\Greekmath 010C} }_{j}
\end{array}
\right) +\mathbf{p}_{nT},
\end{equation*}
where
\begin{equation*}
\mathbf{p}_{nT}=n^{-1/2}\left(
\begin{array}{c}
2\mathbf{{\Greekmath 010C} }_{j}^{\prime }\sqrt{n}\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}
\sqrt{n}\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) +\sqrt{n}
\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) ^{\prime }\mathbf{
Q}_{\bar{w}\bar{w}}\sqrt{n}\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }
_{j}\right) \\
\sqrt{n}\left( \hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) \sqrt{n}\left( \mathbf{
\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) -\sqrt{n}\mathbf{\tilde{Q}}_{
\bar{w}\bar{w}}\sqrt{n}\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }
_{j}\right)
\end{array}
\right) +O(n^{1/2}T^{-2})\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
It then follows that $\mathbf{p}_{nT}$ is of lower order as compared to$
\sqrt{n}\left( \hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) $ and $\sqrt{n}\left(
\mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right) $, and as $
n,T\rightarrow \infty $ jointly, such that $T\approx n^{d}$ for $d>1/2$, we
have
\begin{equation*}
\mathbf{\Omega }_{j}\left(
\begin{array}{c}
\sqrt{n}\left( \hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) \\
\sqrt{n}\left( \mathbf{\hat{{\Greekmath 010C}}}_{j}-\mathbf{{\Greekmath 010C} }_{j}\right)
\end{array}
\right) =\left(
\begin{array}{c}
\sqrt{n}\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}
\mathbf{{\Greekmath 010C} }_{j} \\
-\sqrt{n}\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}\mathbf{{\Greekmath 010C} }_{j}
\end{array}
\right) +o_{p}(1),
\end{equation*}
where $\mathbf{\Omega }_{j}=\left(
\begin{array}{cc}
1 & -2{\Greekmath 0115} _{j}\mathbf{{\Greekmath 010C} }_{j}^{\prime } \\
-\mathbf{{\Greekmath 010C} }_{j} & E\left( \mathbf{Q}_{\bar{w}\bar{w}}\right) -{\Greekmath 0115}
_{j}\mathbf{I}_{m}
\end{array}
\right) $. Using partitioned inverse it is easily seen that $\mathbf{\Omega }
_{j}$ has an inverse if $\mathbf{\Upsilon }_{j}=E\left( \mathbf{Q}_{\bar{w}
\bar{w}}\right) -{\Greekmath 0115} _{j}\mathbf{I}_{m}-2{\Greekmath 0115} _{j}\mathbf{{\Greekmath 010C} }_{j}
\mathbf{{\Greekmath 010C} }_{j}^{\prime }$ is invertible. To check the invertibility of $
\mathbf{\Upsilon }_{j}$ we note that $\mathbf{\Upsilon }_{j}\mathbf{{\Greekmath 010C} }
_{j}=-2{\Greekmath 0115} _{j}\mathbf{{\Greekmath 010C} }_{j}+O\left( T^{-2}\right) $, and since $
{\Greekmath 0115} _{j}>0$ for $j>r_{0}$ it then follows that $\mathbf{\Upsilon }_{j}$
must be invertible. Therefore, we can now solve for $\sqrt{n}\left( \hat{
{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) $, in terms of a linear combination of$
\sqrt{n}\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}
\mathbf{{\Greekmath 010C} }_{j}$ and $\sqrt{n}\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}\mathbf{
{\Greekmath 010C} }_{j}$, and its asymptotic distribution can be derived accordingly.
Consider the asymptotic distribution of these two terms, and note that since
$\mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}=T^{-1}q^{-1}\sum_{\ell =1}^{q}\left(
\mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) \left( \mathbf{
\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) ^{\prime }$, then
\begin{equation*}
\sqrt{n}\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}
\mathbf{{\Greekmath 010C} }_{j}=n^{-1/2}\sum_{i=1}^{n}\mathbf{{\Greekmath 010C} }_{j}^{\prime }
\mathbf{C}_{i}\left[ \mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}-E\left( \mathbf{Q}_{
\bar{s}_{i}\bar{s}_{i}}\right) \right] \mathbf{C}_{i}^{\prime }\mathbf{{\Greekmath 010C}
}_{j}=n^{-1/2}\sum_{i=1}^{n}q^{-1}\sum_{\ell =1}^{q}\left[ {\Greekmath 0110} _{ij,\ell
}^{2}-E\left( {\Greekmath 0110} _{ij,\ell }^{2}\right) \right] ,
\end{equation*}
where$\ {\Greekmath 0110} _{ij,\ell }=T^{-1/2}\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{C}
_{i}\left( \mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) $.
Also by Minkowski inequality
\begin{equation*}
\left( E\left\vert {\Greekmath 0110} _{ij,\ell }\right\vert ^{4+{\Greekmath 010F} }\right)
^{1/4+{\Greekmath 010F} }\leq \left\Vert \mathbf{C}_{i}\right\Vert \left( \left[
E\left\Vert T^{-1/2}\mathbf{\bar{s}}_{i\ell }\right\Vert ^{4+{\Greekmath 010F} }
\right] ^{1/4+{\Greekmath 010F} }+\left[ E\left\Vert T^{-1/2}\mathbf{\bar{s}}_{i\ell
}\right\Vert ^{4+{\Greekmath 010F} }\right] ^{1/4+{\Greekmath 010F} }\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
Using Lemma \ref{Llc} in the supplement we have $\sup_{i,\ell }\left\Vert
T^{-1/2}\mathbf{\bar{s}}_{i\ell }\right\Vert ^{4+{\Greekmath 010F} }<K$, and hence $
\left( E\left\vert {\Greekmath 0110} _{ij,\ell }\right\vert ^{4+{\Greekmath 010F} }\right)
^{1/4+{\Greekmath 010F} }<K$ and $\sqrt{n}\mathbf{{\Greekmath 010C} }_{j}^{\prime }\mathbf{\tilde{
Q}}_{\bar{w}\bar{w}}\mathbf{{\Greekmath 010C} }_{j}\rightarrow _{d}N(0,{\Greekmath 0124} _{j}^{2})$
, for some ${\Greekmath 0124} _{j}^{2}>0$. Similarly, it follows that all $m$
elements\thinspace of $\sqrt{n}\mathbf{\tilde{Q}}_{\bar{w}\bar{w}}\mathbf{
{\Greekmath 010C} }_{j}=n^{-1/2}\sum_{i=1}^{n}\mathbf{C}_{i}\left[ \mathbf{Q}_{\bar{s}
_{i}\bar{s}_{i}}-E\left( \mathbf{Q}_{\bar{s}_{i}\bar{s}_{i}}\right) \right]
\mathbf{C}_{i}^{\prime }\mathbf{{\Greekmath 010C} }_{j}$ are asymptotically normally
distributed with zero means and finite variances. Therefore, it also follows
that $\sqrt{n}\left( \hat{{\Greekmath 0115}}_{j}-{\Greekmath 0115} _{j}\right) \overset{a}{
\thicksim }N(0,{\Greekmath 0124} _{{\Greekmath 0115} _{j}}^{2})$, for some ${\Greekmath 0124} _{{\Greekmath 0115}
_{j}}^{2}>0$. Using this result and setting $j=r_{0}+1$ we have
\begin{equation*}
\Pr \left( \hat{{\Greekmath 0115}}_{r_{0}+1}<C_{T}\right) \overset{a}{\thicksim }
\,\Phi \left[ \frac{\sqrt{n}}{{\Greekmath 0124} _{{\Greekmath 0115} _{r_{0}+1}}}\left(
C_{T}-{\Greekmath 0115} _{r_{0}+1}\right) \right] =\Phi \left[ \frac{-{\Greekmath 0115}
_{r_{0}+1}\sqrt{n}}{{\Greekmath 0124} _{{\Greekmath 0115} _{r_{0}+1}}}\left( 1-\frac{C_{T}}{
{\Greekmath 0115} _{r_{0}+1}}\right) \right] .
\end{equation*}
Since ${\Greekmath 0115} _{r_{0}+1}>0$, then $\Pr \left( \hat{{\Greekmath 0115}}
_{r_{0}+1}<C_{T}\right) \rightarrow 0$, as $n$ $\rightarrow \infty $ if $
C_{T}<{\Greekmath 0115} _{r_{0}+1}$. Recall also from (\ref{PrL1}) that $\Pr \left(
\hat{{\Greekmath 0115}}_{1}\geq C_{T}\right) =O\left( C_{T}^{-1}n^{-1/2}T^{-2}\right) $
, and $\Pr \left( \hat{{\Greekmath 0115}}_{1}\geq C_{T}\right) \rightarrow 0$ if $
C_{T}^{-1}$ does not rise too fast with $T$. Setting $T=KT^{-{\Greekmath 010E} }$, and
recalling that $n\approx T^{1/d}$, these two conditions on $C_{T}$ are met
if $0<{\Greekmath 010E} <2+(1/2d)$. Using this result in (\ref{MSErhat}) and (\ref
{Prrhat}), it follows that $\hat{r}\rightarrow _{p}r_{0}$ and $E\left( \hat{r
}-r_{0}\right) ^{2}\rightarrow 0$, as $n$ and $T\rightarrow \infty $, if $
{\Greekmath 010E} $ is set close to zero such that $KT^{-{\Greekmath 010E} }<$ ${\Greekmath 0115} _{r_{0}+1}$
. \ These results also suggest that the probability of selecting too many
long-run relations is more affected by $n$ than $T$, and the probability of
selecting too few long-run relations tends to zero much faster with $T$ than
with $n$. We need large $n$ for not selecting more than $r_{0}$ long-run
relations. This latter probability, $\Pr \left( \hat{{\Greekmath 0115}}
_{j}<C_{T}\right) $ for $j>r_{0}$, is also affected by the size of ${\Greekmath 0115}
_{r_{0}+1}$ which measures the degree to which there is a transition from a
stationary linear combination under\ which ${\Greekmath 0115} _{j}=0$ for $j\leq r_{0}$
, to ${\Greekmath 0115} _{j}>0$ for ~$j>r_{0}$.

Our theoretical derivations also provide some insight on how to set $K$ and $
{\Greekmath 010E} $ when choosing $C_{T}$. It is clear that ${\Greekmath 010E} $ need not be too
large, so long as $K\thickapprox {\Greekmath 0115} _{r_{0}+1}$. In practice, this can
be achieved approximately, by appropriate scaling of the observations, $
\mathbf{w}_{it}$, as discussed below.

\begin{remark}
The above derivations also suggest that our proposed selection/estimation
procedure would be valid even if there were near stationary relations,
namely if ${\Greekmath 0115} _{r_{0}+1}\thickapprox n^{-b}$ for some $b>0$. Then $\Pr
\left( \hat{{\Greekmath 0115}}_{r_{0}+1}<C_{T}\right) \rightarrow 0,$ so long as $
b<1/2 $, which could be viewed as local-to-zero eigenvalue. Such cases will
not be pursued in this paper, where we require ${\Greekmath 0115} _{r_{0}+1}>0$.
\end{remark}

\section{Allowing for interactive time effects\label{IntEffects}}

The model with interactive time effects is given by (\ref{Grep}), which we
reproduce here for convenience:
\begin{equation}
\mathbf{w}_{it}=\mathbf{a}_{i}+\mathbf{G}_{i}\mathbf{f}_{t}+\mathbf{C}_{i}
\mathbf{s}_{it}+\mathbf{v}_{it},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }t=1,2,...,T;i=1,2,...,n,
\label{gmf}
\end{equation}
where $\mathbf{f}_{t}$ is an $m_{f}\times 1$ vector of latent factors with $
\mathbf{G}_{i}$ the associated $m\times m_{f}$ matrix of factor loadings. We
assume $\mathbf{f}_{t}$ is covariance stationary, and treat the factor
loadings, $\mathbf{G}_{i}$,\ as nonstochastic, without placing any
restrictions on them, besides being uniformly bounded. Hence, latent factors
could be strong, semi-strong or weak. However, we do not allow for the
possibility of unit roots in latent factors, and therefore do not consider
the case of cointegration between $\mathbf{w}_{it}$ and latent factors.
Specifically, we assume:

\begin{assumption}
\label{ASSfactors}(i) The $m_{f}\times 1$ vector of latent factors, $\mathbf{
f}_{t}$, is given by $\mathbf{f}_{t}=\mathbf{\Phi }_{f}(L)\mathbf{
{\Greekmath 0122} }_{ft}=\sum_{j=0}^{\infty }\mathbf{\Phi }_{f\ell }L^{\ell }
\mathbf{{\Greekmath 0122} }_{f,t-\ell }$, for $t=...0,1,2,...,T$, where $\mathbf{
{\Greekmath 0122} }_{ft}$ is an $m_{f}\times 1$ vector of errors distributed
independently over $t$ with $E(\mathbf{{\Greekmath 0122} }_{ft})=\mathbf{0,}$ and $
\sup_{it}E\left\Vert \mathbf{{\Greekmath 0122} }_{ft}\right\Vert ^{4+{\Greekmath 010F} }<K$
, for some ${\Greekmath 010F} >0$. $\mathbf{{\Greekmath 0122} }_{ft}$ is independently
distributed of $\mathbf{u}_{it^{\prime }}$, for all $i,t,t^{\prime }$. (ii)
The $m\times m_{f}$ coefficient matrices $\mathbf{\Phi }_{f\ell }$ are
non-stochastic constants such that $\left\Vert \mathbf{\Phi }_{f\ell
}\right\Vert <K{\Greekmath 011A} ^{\ell }$, where ${\Greekmath 011A} $ lies in the range $0<{\Greekmath 011A} <1$.
(iii) $\mathbf{G}_{i}\,\ $are nonstochastic constants such that $
\sup_{i}\left\Vert \mathbf{G}_{i}\right\Vert <K$.
\end{assumption}

Under Assumption \ref{ASSfactors}, $E\left( \mathbf{G}_{i}\mathbf{f}
_{t}\right) $ is time invariant, and together with Assumption \ref{ASS1}, $
E\left( \mathbf{w}_{it}\right) $ continues to be time invariant. Subtracting
sub-sample time averages from the full sample time average, $\mathbf{\bar{w}}
_{i\ell }-\mathbf{\bar{w}}_{i\circ }$, will therefore continue to remove
unit-specific means.\ More specifically, under (\ref{gmf}) $\mathbf{Q}_{\bar{
w}_{i}\bar{w}_{i}}$ given by (\ref{Qwiwi}) has the following expension
\begin{equation}
\mathbf{Q}_{\bar{w}_{i}\bar{w}_{i}}=\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}
\bar{s}_{i}}\mathbf{C}_{i}^{\prime }+\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}
\bar{v}_{i}}+\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{f}_{i}}+\mathbf{Q}_{
\bar{v}_{i}\bar{s}_{i}}^{\prime }\mathbf{C}_{i}^{\prime }+\mathbf{Q}_{\bar{f}
_{i}\bar{s}_{i}}^{\prime }\mathbf{C}_{i}+\mathbf{Q}_{\bar{f}_{i}\bar{f}_{i}}+
\mathbf{Q}_{\bar{f}_{i}\bar{v}_{i}}+\mathbf{Q}_{\bar{v}_{i}\bar{f}_{i}}+
\mathbf{Q}_{\bar{v}_{i}\bar{v}_{i}},  \label{Qwiwi_f}
\end{equation}
where $\mathbf{Q}_{\bar{f}_{i}\bar{s}_{i}}=\left( Tq\right) ^{-1}\sum_{\ell
=1}^{q}\mathbf{G}_{i}\left( \mathbf{\bar{f}}_{\ell }-\mathbf{\bar{f}}_{\circ
}\right) \left( \mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right)
^{\prime }=\mathbf{Q}_{\bar{s}_{i}\bar{f}_{i}}^{\prime }$,
\begin{equation*}
\mathbf{Q}_{\bar{f}_{i}\bar{v}_{i}}=\left( Tq\right) ^{-1}\sum_{\ell =1}^{q}
\mathbf{G}_{i}\left( \mathbf{\bar{f}}_{\ell }-\mathbf{\bar{f}}_{\circ
}\right) \left( \mathbf{\bar{v}}_{i\ell }-\mathbf{\bar{v}}_{i\circ }\right)
^{\prime }=\mathbf{Q}_{\bar{v}_{i}\bar{f}_{i}}^{\prime }
\end{equation*}
$\mathbf{Q}_{\bar{f}_{i}\bar{f}_{i}}=\left( Tq\right) ^{-1}\sum_{\ell =1}^{q}
\mathbf{G}_{i}\left( \mathbf{\bar{f}}_{\ell }-\mathbf{\bar{f}}_{\circ
}\right) \left( \mathbf{\bar{f}}_{\ell }-\mathbf{\bar{f}}_{\circ }\right)
^{\prime }\mathbf{G}_{i}^{\prime }$, and the terms not involving the latent
factor are as before. By Lemma \ref{LQB} in the supplement we have
\begin{equation}
\sup_{i}E\left\Vert \mathbf{Q}_{\bar{f}_{i}\bar{f}_{i}}\right\Vert =O\left(
T^{-2}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\sup_{i}E\left\Vert \mathbf{Q}_{\bar{f}_{i}\bar{v}
_{i}}\right\Vert =O\left( T^{-2}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }\sup_{i}E\left\Vert
\mathbf{Q}_{\bar{f}_{i}\bar{s}_{i}}\right\Vert =O\left( T^{-1}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.
}  \label{Order_fterms}
\end{equation}
Pooling over $i$, $\mathbf{Q}_{\bar{w}\bar{w}}=n^{-1}\sum_{i=1}^{n}\mathbf{Q}
_{\bar{w}_{i}\bar{w}_{i}}$, and using the above results together with those
already established in (\ref{supQsv}) and (\ref{CQss}) now yields
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}=\frac{(q-1)}{6q}\mathbf{\Psi }_{n}+O_{p}\left(
n^{-1/2}\right) +O_{p}\left( T^{-1}\right) ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }\mathbf{Q}_{\bar{w}
\bar{w}}\mathbf{{\Greekmath 010C} }_{j0}=O_{p}\left( n^{-1/2}\right) +O_{p}\left(
T^{-1}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}  \label{Qwwfo}
\end{equation}
which is identical to results (\ref{Qbar2})-(\ref{QbarBeta}) in Proposition
\ref{Qwbar} for models without interactive time effects.\textbf{\ }
Similarly, inclusion of interactive time effects does not alter the
convergence rate of the exactly identified PME estimator $\widehat{\mathbf{
\mathring{B}}}_{0}$, given by (\ref{BETAdot_hat}), and its asymptotic
distribution will remain correctly centered at zero. To see this consider
the following expression for $\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left(
\widehat{\mathbf{\mathring{B}}}_{0}-\mathbf{\mathring{B}}_{0}\right) $,
which extends (\ref{qwb}) to panel data models with interactive time
effects,
\begin{eqnarray*}
\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left( \widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}\right) &=&-\left( n^{-1/2}\sum_{i=1}^{n}T
\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}\right) \mathbf{\mathring{B}
}_{0}-\left( n^{-1/2}\sum_{i=1}^{n}T\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}
\bar{f}_{i}}\right) \mathbf{\mathring{B}}_{0} \\
&&-\frac{\sqrt{n}}{T}\left( n^{-1}\sum_{i=1}^{n}T^{2}\mathbf{Q}_{\bar{f}_{i}
\bar{f}_{i}}\right) \mathbf{\mathring{B}}_{0}-\frac{\sqrt{n}}{T}\left[
n^{-1}\sum_{i=1}^{n}T^{2}\left( \mathbf{Q}_{\bar{f}_{i}\bar{v}_{i}}+\mathbf{Q
}_{\bar{v}_{i}\bar{f}_{i}}\right) \right] \mathbf{\mathring{B}}_{0} \\
&&-\frac{\sqrt{n}}{T}\left( n^{-1}\sum_{i=1}^{n}T^{2}\mathbf{Q}_{\bar{v}_{i}
\bar{v}_{i}}\right) \mathbf{\mathring{B}}_{0}+O_{p}\left( T^{-1}\right)
\mathbf{.}
\end{eqnarray*}
The new terms involve matrices $\mathbf{Q}_{\bar{s}_{i}\bar{f}_{i}}$, $
\mathbf{Q}_{\bar{f}_{i}\bar{f}_{i}}$ and $\mathbf{Q}_{\bar{f}_{i}\bar{v}
_{i}}=\mathbf{Q}_{\bar{v}_{i}\bar{f}_{i}}^{\prime }$. Using the bounds in (
\ref{Order_fterms}), we obtain $E\left\Vert n^{-1}\sum_{i=1}^{n}T^{2}\mathbf{
Q}_{\bar{f}_{i}\bar{f}_{i}}\right\Vert \leq
n^{-1}\sum_{i=1}^{n}T^{2}E\left\Vert \mathbf{Q}_{\bar{f}_{i}\bar{f}
_{i}}\right\Vert =O\left( T^{-2}\right) $, and similarly \newline
$E\left\Vert n^{-1}\sum_{i=1}^{n}T^{2}\left( \mathbf{Q}_{\bar{f}_{i}\bar{v}
_{i}}+\mathbf{Q}_{\bar{v}_{i}\bar{f}_{i}{}_{i}}\right) \right\Vert <K$. In
addition, by Lemma \ref{L_sqvvs} recall that $n^{-1}\sum_{i=1}^{n}T^{2}
\mathbf{Q}_{\bar{v}_{i}\bar{v}_{i}}=O_{p}\left( n^{-1/2}\right) $. Using
these results and noting that $\left\Vert \mathbf{\mathring{B}}
_{0}\right\Vert <K$, we have
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left( \widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}\right) =-\left( n^{-1/2}\sum_{i=1}^{n}T
\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}\right) \mathbf{\mathring{B}
}_{0}-\left( n^{-1/2}\sum_{i=1}^{n}T\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}
\bar{f}_{i}}\right) \mathbf{\mathring{B}}_{0}+O_{p}\left( \frac{\sqrt{n}}{T}
\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}  \label{qbf}
\end{equation}
To simplify the notations we set $\mathbf{{\Greekmath 0121} }_{it}=\mathbf{f}_{it}+
\mathbf{v}_{it}$, and note that
\begin{equation*}
\mathbf{Q}_{\bar{s}_{i}\bar{{\Greekmath 0121}}_{i}}=\mathbf{Q}_{\bar{s}_{i}\bar{v}_{i}}+
\mathbf{Q}_{\bar{s}_{i}\bar{f}_{i}}=T^{-1}q^{-1}\sum_{\ell =1}^{q}\left(
\mathbf{\bar{s}}_{i\ell }-\mathbf{\bar{s}}_{i\circ }\right) \left( \mathbf{
\bar{{\Greekmath 0121}}}_{i\ell }-\mathbf{\bar{{\Greekmath 0121}}}_{i\circ }\right) ^{\prime }
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
Then (\ref{qbf}) can be written as
\begin{equation*}
\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left( \widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}\right) =-\left( n^{-1/2}\sum_{i=1}^{n}T
\mathbf{C}_{i}\mathbf{Q}_{\bar{s}_{i}\bar{{\Greekmath 0121}}_{i}}\right) \mathbf{
\mathring{B}}_{0}+O_{p}\left( \frac{\sqrt{n}}{T}\right) .
\end{equation*}
Let $\mathbf{Z}_{i}^{\ast }=\mathbf{C}_{i}\left[ T\mathbf{Q}_{\bar{s}_{i}
\bar{{\Greekmath 0121}}_{i}}-E\left( T\mathbf{Q}_{\bar{s}_{i}\bar{{\Greekmath 0121}}_{i}}\right)
\right] \mathbf{\mathring{B}}_{0}$. Since $\mathbf{s}_{it}$ is independent
of $\mathbf{f}_{t}$ and $E\left( \mathbf{s}_{it}\right) =\mathbf{0}$, we
have $E\left( \mathbf{Q}_{\bar{s}_{i}\bar{f}_{i}}\right) =\mathbf{0}$, and
using (\ref{T2qsvb}) we obtain $\left\Vert n^{-1}\sum_{i=1}^{n}\mathbf{C}
_{i}E\left( T^{2}\mathbf{Q}_{\bar{s}_{i}\bar{{\Greekmath 0121}}_{i}}\right) \mathbf{
{\Greekmath 010C} }_{0}\right\Vert <K$. It now follows that
\begin{equation}
\mathbf{Q}_{\bar{w}\bar{w}}\sqrt{n}T\left( \widehat{\mathbf{\mathring{B}}}
_{0}-\mathbf{\mathring{B}}_{0}\right) =-n^{-1/2}\sum_{i=1}^{n}\mathbf{Z}
_{i}^{\ast }+O_{p}\left( \frac{\sqrt{n}}{T}\right) ,  \label{qbz}
\end{equation}
where $\mathbf{Z}_{i}^{\ast }=\mathbf{C}_{i}\left[ T\mathbf{Q}_{\bar{s}_{i}
\bar{{\Greekmath 0121}}_{i}}-E\left( T\mathbf{Q}_{\bar{s}_{i}\bar{{\Greekmath 0121}}_{i}}\right)
\right] \mathbf{{\Greekmath 010C} }_{0}$. Writing (\ref{qbz}) in vec form
\begin{equation*}
\left( \mathbf{I}_{r}\mathbf{\otimes Q}_{\bar{w}\bar{w}}\right) \sqrt{n}T
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }vec\left( \widehat{\mathbf{\mathring{B}}}_{0}-\mathbf{\mathring{B}}
_{0}\right) =n^{-1/2}\sum_{i=1}^{n}\left( \mathbf{\mathring{B}}_{0}^{\prime }
\mathbf{\otimes \mathbf{C}}_{i}\right) \left[ \mathbf{{\Greekmath 0118} }_{iq}^{\ast
}-E\left( \mathbf{{\Greekmath 0118} }_{iq}^{\ast }\right) \right] +O_{p}\left( \frac{\sqrt{
n}}{T}\right) ,
\end{equation*}
where $\mathbf{{\Greekmath 0118} }_{iq}^{\ast }$ is the $m^{2}\times 1$ vector given by $
\mathbf{{\Greekmath 0118} }_{iq}^{\ast }=q^{-1}\sum_{s=1}^{q}\left( \mathbf{\bar{{\Greekmath 0121}}}
_{is}-\mathbf{\bar{{\Greekmath 0121}}}_{i}\right) \mathbf{\otimes }\left( \mathbf{\bar{s
}}_{is}-\mathbf{\bar{s}}_{i}\right) $. Although $\mathbf{{\Greekmath 0118} }_{iq}^{\ast }$
is not independent over $i$, it is a martingale difference sequence, and
\newline
$n^{-1/2}\sum_{i=1}^{n}\left( \mathbf{\mathring{B}}_{0}^{\prime }\mathbf{
\otimes \mathbf{C}}_{i}\right) \left[ \mathbf{{\Greekmath 0118} }_{iq}^{\ast }-E\left(
\mathbf{{\Greekmath 0118} }_{iq}^{\ast }\right) \right] $ converges to a normal
distribution as $n,T\rightarrow \infty $, jointly. Therefore, $\widehat{
\mathbf{\mathring{B}}}$ will continue to be asymptotically normally
distributed, with its asymptotic distribution correctly centered at zero.
Even though the variance of the asymptotic distribution of PME estimator in
general depends on the interactive time effects, the estimator of the
asymptotic variance given by (\ref{VarHatTheta}) will continue to be
consistent under Assumption \ref{ASSfactors}, and inference can be carried
out in the same way as in panel data models without interactive time effects.

Overall, we find that the PME estimator is robust to interactive time
effects so long as Assumption \ref{ASSfactors} holds. Theorems and \ref
{Tconsf} and \ref{Tbf} in the supplement provide formal statements regarding
consistency and the asymptotic normality of the PME estimators in presence
of interactive time effects.

\section{How to choose $q$ and $C_{T}$\label{qCTchoices}}

To implement the estimation of $r_{0}$ and the associated long-run relations
we need to decide on $q$, and $C_{T}=CT^{-{\Greekmath 010E} }$ that enter the
thresholding estimation of $r_{0}$.

\subsection{Choice of $q$\label{qchoice}}

To ensure that $\mathbf{Q}_{\bar{w}\bar{w}}=n^{-1}T^{-1}q^{-1}\sum_{i=1}^{n}
\sum_{\ell =1}^{q}\left( \mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ
}\right) \left( \mathbf{\bar{w}}_{i\ell }-\mathbf{\bar{w}}_{i\circ }\right)
^{\prime }$ does not depend on the fixed effects, given by $E(\mathbf{w}
_{it})=\mathbf{a}_{i}$, we need at least two sub-samples, namely $q\geq 2$.
To reduce the variance of time series dependence of $\mathbf{\bar{w}}_{i\ell
}-\mathbf{\bar{w}}_{i\circ }$ over the sub-samples we need $T/q$ to be
reasonably large. An optimum choice of $q$ most likely depends on $T$, and
not so much on $n$. To ensure that the mathematical derivations are
manageable and transparent, so far we have assumed that $q$ is fixed as $
T\rightarrow \infty $. But to select $q$ we must allow $q$ to depend on $T$,
denoted as $q_{T}$, and consider values of $q_{T}$ that rise with $T$, but
at a slower rate such that $q_{T}/T\rightarrow 0$. Analogous to the problem
of selecting the lag order in time series literature, we conjecture setting $
q_{T}$ to rise at the rate of $T^{1/3}$, and set $q_{T}$ to the lower
integer part of $\max (2,T^{1/3})$. This would suggest that values of $q_{T}$
equal to $2$, $3$ and $4$ for values of $T=20,50$ and $100$, respectively,
considered in our Monte Carlo simulations reported in Section \ref{MC}
below, where we consider the values of $2$ and $4$, to save space. For
estimation of $r_{0}$, the choice of $q=2$ works perfectly well for all
values of $T$ considered. But the higher value of $q=4$ does seem to perform
slightly better than $q=2$ for estimation of long-run coefficients when $
T=100$.

\subsection{Choices of $C$ and $\protect{\Greekmath 010E} $ for estimation of $r_{0}$}

It is clear that eigenvalues of $\mathbf{Q}_{\bar{w}\bar{w}}$ depend on the
scale of the observations, $\mathbf{w}_{it}$, and some form of scaling of
data is required to reduce the sensitivity of the eigenvalues to scale. One
could resort to cross validation procedures to set $C$ and ${\Greekmath 010E} $, but
based on extensive Monte Carlo experiments, we have found that setting $C=1$
works well if we base our selection procedure on the eigenvalues of the
following correlation matrix
\begin{equation}
\mathbf{R}_{_{\bar{w}\bar{w}}}=\left[ diag\left( \mathbf{Q}_{\bar{w}\bar{w}
}\right) \right] ^{-1/2}\mathbf{Q}_{\bar{w}\bar{w}}\left[ diag\left( \mathbf{
Q}_{\bar{w}\bar{w}}\right) \right] ^{-1/2}.  \label{Rww}
\end{equation}
Accordingly, our proposed estimator of $r_{0}$ is given by
\begin{equation}
\tilde{r}=\sum_{j=1}^{m}\mathcal{I}\left( \tilde{{\Greekmath 0115}}_{j}<T^{-{\Greekmath 010E}
}\right) ,  \label{r_est}
\end{equation}
where $\tilde{{\Greekmath 0115}}_{j}$, for $j=1,2,...,m$ are the eigenvalues of $
\mathbf{R}_{_{\bar{w}\bar{w}}}$.\footnote{
Using the correlation matrix, $\mathbf{R}_{_{\bar{w}\bar{w}}}$, has the
advantage that it is unaffected by scaling, so long as the same scaling is
used across all cross section units. It is not invariant if the scaling
varies across units as well as across the variables. This issues is
addressed in Section \ref{Sup_MCs} of the supplement where we investigate
the small sample sensitivity of $\tilde{r}$ to differential scaling of the
variables across the units. We find that the small sample performance of $
\tilde{r}$ as an estimator of $r_{0}$, is hardly affected by such
differential scaling.}

Also, based on our theoretical derivations any value of ${\Greekmath 010E} $ close to $
zero$ should work. In the Monte Carlo experiments we consider the values of $
{\Greekmath 010E} =1/4$ and $1/2$ and conclude that ${\Greekmath 010E} =1/4$ is a good overall
choice and cross-validation is not necessary for the implementation of our
estimation strategy.

\section{Monte Carlo Evidence\label{MC}}

We investigate small sample properties of the proposed PME estimator with
Monte Carlo experiments using both VARMA(1,1) and VAR(1) designs, with and
without interactive time effects. The designs we consider are all special
cases of the general linear model (\ref{Grep}). We set $m=3$, and generate
the $3\times 1$ vector $\mathbf{w}_{it}$ as $I(1)$ variables under three
scenarios: non-cointegration, $r_{0}=0$, and $r_{0}=1$ and $r_{0}=2$
cointegrating relations. We consider sample size combinations, $
T=(20,50,100) $ and $n=(50,500,1000,3000)$, and report results for $q=2$ and
$q=4$ (sub-samples), which are in line with our conjecture of setting $q$ in
line with the $\max (2,T^{1/3})$ rule. See Section \ref{qchoice}.

When $r_{0}=1$, there are a range of alternative estimators of the
cointegrating relation in the literature that we can use for comparison,
most of which assume the direction of long-run causality is known. To
accommodate existing estimators, we distinguish between experiments based on
long-run causal ordering. The PME estimator does not require the direction
of long-run causality to be known. When $r_{0}>1$, to the best of our
knowledge, there are no obvious alternative estimators of $r_{0}$ and the
associated cointegrating relations in the panel cointegration literature
that we can use. Accordingly, for the purpose of comparison, we report
results using a mean group version of Johansen's maximum likelihood
procedure, whereby we estimate $r_{0}$ and associated cointegrating vectors
(if any), for all individual units in the panel separately, and report the
frequency with which $r_{0}$ is selected by Johansen procedure across the $n$
units, and the mean group estimates of the cointegrating coefficients and
their standard errors.

Subsection \ref{DGPsubsection} outlines the data generating processes
(DGPs). Subsection \ref{number} gives the results for estimates of $r_{0}$.
Our results show near perfect performance for our proposed estimator of $
r_{0}$, even for samples as small as $T=20$ and $n=50$. This is in line with
the theory developed in Section \ref{r_sel}. Subsection \ref{coefficients}
reports the results for the coefficients of the long-run relations assuming $
r_{0}$ is known, which is justified considering the near perfect performance
of our estimator of $r_{0}$. The MC results provide simulation evidence that
the PME estimator performs well in panels with $n$ as large as $1,000$ and $
T $ as small as $20$.

\subsection{Data generating processes\label{DGPsubsection}}

We consider experiments with and without long-run relations. In the
experiments with long-run relations, $\mathbf{w}_{it}$ is generated as
\begin{equation}
\Delta \mathbf{w}_{it}=\mathbf{d}_{i}-\mathbf{\Pi }_{i}\mathbf{w}_{i,t-1}+
\mathbf{u}_{it}-\mathbf{\Theta }_{i}\mathbf{u}_{i,t-1},  \label{varma}
\end{equation}
for $i=1,2,...,n$, $t=1,2,...,T$, where $\mathbf{\Pi }_{i}=\mathbf{A}_{i}
\mathbf{B}_{0}^{\prime },$ $\mathbf{A}_{i}$ is $m\times r_{0},$ $\mathbf{B}
_{0}$ is $m\times r_{0}.$ We set $\mathbf{d}_{i}=\mathbf{\Pi }_{i}\mathbf{
{\Greekmath 0116} }_{iw}$ to ensure no linear trends in data. See, for example, Section
5.7 in
\citeN{Johansen1995}
.\textbf{\ }The elements of $\mathbf{{\Greekmath 0116} }_{iw}$ are generated as $IIDN(0,1)$
. We consider both VAR(1) and VARMA(1,1) designs. For VAR(1) we set $\mathbf{
\Theta }_{i}=\mathbf{0}$, and for VARMA(1,1) with set $\mathbf{\Theta }
_{i}=diag({\Greekmath 0112} _{ij},j=1,2,...,m)$, and generate ${\Greekmath 0112} _{ij}$ as $IIDU
\left[ -0.5,0.5\right] $. The errors $\mathbf{u}_{it}$, are generated
following both Gaussian and chi-squared distributions.

We initially consider $m=3$ variables in $\mathbf{w}_{it}=\left(
w_{it,1},w_{it,2},w_{it,3}\right) ^{\prime }$, with both one and two
long-run relations. For $r_{0}=1$, we set $\mathbf{B}_{0}=\mathbf{{\Greekmath 010C} }
_{1,0}=\left( 1,0,-1\right) ^{\prime }$ and $\mathbf{A}
_{i}=(a_{i,11},a_{i,21},...,a_{i,31})^{\prime }$. In this case, the long-run
relation is given by
\begin{equation}
\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }\mathbf{w}_{it}=w_{it,1}-w_{it,3}={\Greekmath 010C}
_{11,0}w_{it,1}+{\Greekmath 010C} _{12,0}w_{it,2}+{\Greekmath 010C} _{13,0}w_{it,3}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\label{r1}
\end{equation}
with ${\Greekmath 010C} _{11,0}=1$, ${\Greekmath 010C} _{12,0}=0$ and ${\Greekmath 010C} _{13,0}=-1$. We
identify the long-run relation by imposing ${\Greekmath 010C} _{11,0}=1$ and estimate $
{\Greekmath 010C} _{12,0}$ and ${\Greekmath 010C} _{13,0}$. When $r_{0}=2$, we set
\begin{equation*}
\mathbf{B}_{0}=\left( \mathbf{{\Greekmath 010C} }_{1,0},\mathbf{{\Greekmath 010C} }_{2,0}\right)
=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
-1 & -1
\end{array}
\right) ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }\mathbf{A}_{i}=\left(
\begin{array}{cc}
a_{i,11} & a_{i,12} \\
a_{i,21} & a_{i22} \\
a_{i,31} & a_{i,32}
\end{array}
\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
In this case, the two long-run relations are given by
\begin{eqnarray}
\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }\mathbf{w}_{it} &=&w_{it,1}-w_{it,3}={\Greekmath 010C}
_{11,0}w_{it,1}+{\Greekmath 010C} _{12,0}w_{it,2}+{\Greekmath 010C} _{13,0}w_{it,3}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\label{r2a} \\
\mathbf{{\Greekmath 010C} }_{2,0}^{\prime }\mathbf{w}_{it} &=&w_{it,2}-w_{it,3}={\Greekmath 010C}
_{21,0}w_{it,1}+{\Greekmath 010C} _{22,0}w_{it,2}+{\Greekmath 010C} _{23,0}w_{it,3}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\label{r2b}
\end{eqnarray}
where ${\Greekmath 010C} _{11,0}={\Greekmath 010C} _{22,0}=1$, ${\Greekmath 010C} _{12,0}={\Greekmath 010C} _{21,0}=0$, and $
{\Greekmath 010C} _{13,0}={\Greekmath 010C} _{23,0}=-1$. We identify these long-run relations by
imposing ${\Greekmath 010C} _{11,0}={\Greekmath 010C} _{22,0}=1$ and ${\Greekmath 010C} _{12,0}={\Greekmath 010C} _{21,0}=0$
, and we estimate ${\Greekmath 010C} _{13,0}$ and ${\Greekmath 010C} _{23,0}$.

We set the values of $\mathbf{A}_{i}$ to control the average speed of
convergence towards the long-run relations. For example, in the case where $
r_{0}=1$ then $\mathbf{B}_{0}^{\prime }\mathbf{A}_{i}={\Greekmath 011A}
_{i}=a_{i,11}-a_{i,31}$ and ${\Greekmath 011A} _{i}$, for $i=1,2,...,n$ are generated as $
IIDU[0.1,0.2]$ representing slow convergence, and as $IIDU[0.1,0.3]$
representing moderate convergence. We then set $a_{i,21}=0$, which leaves us
with one free parameter in $\mathbf{A}_{i}$ which we use to set the system
measures of the fit, $PR_{nT}^{2}=0.2$ and $0.3,$ defined as a pooled $R^{2}$
, given by equation (\ref{SysPR2}) in the supplement. Since $a_{i,11}$ and $
a_{i,31}$ are both nonzero, the long-run causality runs from $\left(
w_{it,2},w_{it,3}\right) $ to $w_{it,1}$ as well as from $w_{it,1}$ to $
w_{it,3}$. For $r_{0}=2$ the rate of convergence will depend on the
eigenvalues of $\mathbf{I}_{2}-\mathbf{B}_{0}^{\prime }\mathbf{A}_{i}$ and
the details of how we generate the elements of $\mathbf{A}_{i}$ are given in
the supplement.

We also consider experiments with interactive time effects, which are
obtained by augmenting the solution of (\ref{varma}) with $\mathbf{G}_{i}
\mathbf{f}_{t}$, where $\mathbf{f}_{t}$ is and $m_{f}\times 1$ vector of
latent factors. Each of these factors are generated as AR(1) process with a
break in the AR coefficient, and the individual elements of the $m\times
m_{f}$ matrix of factor loadings $\mathbf{G}_{i}$ are generated as $IIDU
\left[ 0.0.4\right] $. We set $m_{f}=4$. Specifically, we augment the
general linear process versions of the above VARMA(1,1) and VAR(1)
specifications with $\mathbf{G}_{i}\mathbf{f}_{t}$\thinspace , namely
\begin{equation*}
\mathbf{w}_{it}=\mathbf{w}_{i0}+\mathbf{G}_{i}\mathbf{f}_{t}+\mathbf{C}_{i}
\mathbf{s}_{it}+\mathbf{C}_{i}^{\ast }(L)\mathbf{u}_{it},
\end{equation*}
where $\mathbf{C}_{i}$ and $\mathbf{C}_{i}^{\ast }(L)$ are obtained from
\begin{equation*}
(\mathbf{I}_{3}-\mathbf{\Psi }_{i}L)\left[ \mathbf{C}_{i}+\mathbf{C}
_{i}^{\ast }(L)(1-L)\right] =(\mathbf{I}_{3}-\mathbf{\Theta }_{i}L)(1-L)
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\end{equation*}
and $\mathbf{\Psi }_{i}=\mathbf{I}_{3}-\mathbf{A}_{i}\mathbf{B}_{0}^{\prime
} $\textbf{. }See the supplement for further details.

For the experiments with no long-run relations ($r_{0}=0$) we generate $
\Delta \mathbf{w}_{it}$ using the VAR(1) model in first-differences:
\begin{equation}
\Delta \mathbf{w}_{it}=\mathbf{\Phi }_{i}\Delta \mathbf{w}_{i,t-1}+\mathbf{u}
_{it},  \label{VARdif}
\end{equation}
for $i=1,2,...,n$, $t=1,2,...,T$, where $\mathbf{u}_{it}\thicksim IIDN(
\mathbf{0},\mathbf{\Sigma }_{ui})$. The elements of the covariance matrix $
\mathbf{\Sigma }_{ui}=\left( {\Greekmath 011B} _{i,\ell \ell ^{\prime }}\right) $ are
generated as ${\Greekmath 011B} _{i,\ell \ell }=1$ for $i=1,2,...,n$ and $\ell =1,2,3$,
and ${\Greekmath 011B} _{i,\ell \ell ^{\prime }}\thicksim IIDU(0,0.5)$, for $\ell \neq
\ell ^{\prime }\,$, and $i=1,2,...,n$. We use a diagonal matrix for $\mathbf{
\Phi }_{i}=({\Greekmath 011E} _{i,\ell \ell ^{\prime }})$, with ${\Greekmath 011E} _{i,\ell \ell }$
elements on its diagonal, for $r=1,2,...,m$. We consider three options for $
{\Greekmath 011E} _{i,\ell \ell }$: ($i$) low values ${\Greekmath 011E} _{i,\ell \ell }\sim U[0,0.8]$,
($ii$) moderate values ${\Greekmath 011E} _{i,\ell \ell }\sim U[0.7,0.9]$, and ($iii$)
high values ${\Greekmath 011E} _{i,\ell \ell }\sim U[0.80,0.95]$. $\mathbf{w}_{it}$ is
then obtained by cumulating $\Delta \mathbf{w}_{it}$ from the initial value $
\mathbf{w}_{i,0}=0$. Similarly to the experiment with long-run relations
given by (\ref{varma}), the model (\ref{VARdif}) is a special case of (\ref
{Grep}). Specifically, (\ref{VARdif}) leads to $\mathbf{w}_{it}=\mathbf{w}
_{i0}+\mathbf{G}_{i}\mathbf{f}_{t}+\mathbf{C}_{i}\mathbf{s}_{it}+\mathbf{C}
_{i}^{\ast }(L)\mathbf{u}_{it},$ where $\mathbf{C}_{i}=\left( \mathbf{I}_{m}-
\mathbf{\Phi }_{i}\right) ^{-1}$, and $\mathbf{C}_{i}^{\ast }(L)=-\mathbf{
\Phi }_{i}\left( \mathbf{I}_{m}-\mathbf{\Phi }_{i}\right) ^{-1}\left(
\mathbf{I}_{m}-\mathbf{\Phi }_{i}L\right) ^{-1}$. For experiments without
interactive time effects we set $\mathbf{G}_{i}=\mathbf{0}$, for all $i$.

In addition to the designs described above, we also consider a data
generating process taken from Section 3.1 of
\citeN{ChudikPesaranSmith2021PB}
. For this $m=2$, and $\mathbf{w}_{it}=\left( w_{1,it},w_{2,it}\right)
^{\prime }$ is generated as
\begin{eqnarray*}
\Delta w_{1,it} &=&c_{i}-a_{i,11}\left( w_{1,i,t-1}-w_{2,i,t-1}\right)
+u_{1,it}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,} \\
\Delta w_{2,it} &=&u_{2,it}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\end{eqnarray*}
where $a_{i,11}\sim IIDU\left[ 0.2,0.3\right] $, $\mathbf{u}_{it}=\left(
u_{1,it},u_{2,it}\right) ^{\prime }$ is heteroskedastic and
cross-sectionally independent, and $u_{1,it}$ is correlated with $u_{2,it}$.
\footnote{$u_{1,it}$ $={\Greekmath 011B} _{1i}e_{1,it}$, $u_{2,it}={\Greekmath 011B} _{2i}e_{2,it}$
, ${\Greekmath 011B} _{1,i}^{2},{\Greekmath 011B} _{2,i}^{2}\sim IIDU\left[ 0.8,1.2\right] $,
\begin{equation*}
\left(
\begin{array}{c}
e_{1,it} \\
e_{2,it}
\end{array}
\right) \sim IIDN\left( \mathbf{0}_{2},\mathbf{\Sigma }_{e}\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }
\mathbf{\Sigma }_{e}\sim \left(
\begin{array}{cc}
1 & {\Greekmath 011A} _{ei} \\
{\Greekmath 011A} _{ei} & 1
\end{array}
\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, and }{\Greekmath 011A} _{ei}\sim IIDU\left[ 0.3,0.7\right] \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
See Section 3.1 of
\citeN{ChudikPesaranSmith2021PB}
for details.} We use this design to see how PME compares to single-equation
estimators that correctly assume long-run causality runs from $w_{2,it}$ to $
w_{1,it}$. We would expect such estimators to perform reasonably well,
particularly when $T$ is large relative to $n$, and provide a good baseline
to evaluate the performance of PME in settings favorable to the single
equation techniques advanced in the literature.

In short, we have 71 experiments. 6 experiments with $m=3$ variables and no
long-run relations ($r_{0}=0$), given by 3 choices for the distribution of $
{\Greekmath 011E} _{ij}$, and interactive time effects are included or not. $64=2^{5}$
experiments feature $m=3$ variables with long-run relations, given by
combinations of the choice of model (VAR(1) or VARMA(1,1)), $r_{0}=1$ or $2$
, Gaussian or chi-square error distributions, moderate or slow speed of
convergence, system measure of fit $PR_{nT}^{2}$ $=0.2$ or $0.3$, and
interactive time effects are included or not. In addition, we have one
experiment with $m=2$ variables, $r_{0}=1$ long-run relation and one one-way
long-run causality taken directly from
\citeN{ChudikPesaranSmith2021PB}
.

To save space, we report only summary results that are averages across a
number of selected experiments, with the results for the individual
experiments available from the authors upon request. Section \ref{Sup_MCd}
of the supplement also provides further details of the Monte Carlo designs,
how the processes are initialized, and the rationale behind the
parameterization adopted. Additionally, Section \ref{Sup_MCr}\ of the
supplement shows that the results for estimation of $r_{0}$ the associated
long-run relations are robust to GARCH and threshold autoregressive effects.

\subsection{Small sample evidence on estimation of $r_{0}$ \label{number}}

We summarize the Monte Carlo findings for the estimation of $r_{0}$ by the
eigenvalue thresholding estimator $\tilde{r}$ given by (\ref{r_est}), using $
{\Greekmath 010E} =1/4$ and $1/2$ in Tables 1-2. Table 1 reports selection frequencies
of $\tilde{r}=0$, $1$, $2$, $3$ using VAR(1) based DGPs without interactive
time effects, for the three cases of $r_{0}=0,1$ or $2$.\footnote{
These results are averaged over 3 VAR(1) experiments in first differences\ ($
r_{0}=0$), differeing in terms of autoregressive coefficients (low, medium
and high values), and over 8 VAR(1) experiments in levels featuring $r_{0}=1$
and $2$ long-run relationships, differing in terms of the error
distributions (Gaussian or chi-squared), fit (high or low), and speed of
convergence towards long run (moderate or low).} The theory indicated very
fast convergence of $\tilde{r}$ to $r_{0}$, and this is confirmed by the
Monte Carlo simulations. The eigenvalue thresholding estimator $\tilde{r}$
correctly identifies $r_{0}$ in $100$ per cent of cases, except for the very
small sample sizes considered. For $n=50$ and $T=20$, we see $5\%$
probability of $\tilde{r}$ overestimating the true number of long-run
relations when the smaller exponent ${\Greekmath 010E} =1/4$ is used in experiments
with $r_{0}=0$ in Table1. For comparison, we also report selection frequency
for the Johansen procedure using the trace statistic and the conventional
nominal level of 5 percent.\footnote{
We assume the true lag order of the VAR design is known.} Whereas there is a
single estimate for the whole panel per replication using $\tilde{r},$ there
are $n$ such estimates per replication using the Johansen procedure ($\hat{r}
_{i}$, $i=1,2,...,n$). We simply use them all in calculating the selection
frequency.\footnote{
There are a number of ways that one might choose $r$ for the panel from
these $n$ tests, based on the modal selection or the average value of the
test statistic for instance. We do not explore these avenues here.} Hence $n$
will not influence these results, but increasing $T$ does improve the
frequency with which the correct value is selected using the trace
statistic. For $T=100$, frequency of correctly estimating the number of
long-run relations using the Johansen trace statistics is $69$ percent for $
r_{0}=0$, $82$ percent for $r_{0}=1$, and only $31$ percent for $r_{0}=2$ in
Table 1.

Simulation results for the performance of $\tilde{r}$ as an estimator of $
r_{0}$ in the case of experiments with interactive time effects are
presented in Section \ref{Sup_MCs} of the supplement, to save space. These
results continue to show near perfect performance of $\tilde{r}$ similarly
to the experiments summarized above for the case of panels without
interactive time effects.

Overall, our findings are in line with the theoretical insights of a very
fast convergence of $\tilde{r}$ to $r_{0}$. For this reason, we report next
on the small sample performance of the identified PME estimator assuming $
r_{0}$ is known.\pagebreak

\begin{center}
\singlespacing
TABLE 1: Selection frequencies, averaged across experiments, for the
estimation of $r_{0}=0,1,2,3$ by eigenvalue thresholding estimator, $\tilde{r
}$, given by (\ref{r_est}) with ${\Greekmath 010E} =1/4$ and $1/2$ and by Johansen's
trace statistics using $VAR(1)$ as the DGP with $r_{0}=0,1,2,$ and without
interactive time effects.\smallskip

\setlength{\tabcolsep}{4pt}
\scriptsize
\begin{tabular}{rrrrrrrrrrrrrrrrr}
\hline\hline
&  & \multicolumn{3}{c}{Frequency $\tilde{r}=0$} &  & \multicolumn{3}{c}{
Frequency $\tilde{r}=1$} &  & \multicolumn{3}{c}{Frequency $\tilde{r}=2$} &
& \multicolumn{3}{c}{Frequency $\tilde{r}=3$} \\
\cline{3-5}\cline{7-9}\cline{11-13}\cline{15-17}
$n$ $\backslash $ $T$ &  & \textbf{20} & \textbf{50} & \textbf{100} &  &
\textbf{20} & \textbf{50} & \textbf{100} &  & \textbf{20} & \textbf{50} &
\textbf{100} &  & \textbf{20} & \textbf{50} & \textbf{100} \\ \hline
\multicolumn{10}{l}{\textbf{A. Experiments with }$r_{0}=0$} &  &  &  &  &  &
&  \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/4$} \\ \hline
\textbf{50} &  & 0.95 & 1.00 & 1.00 &  & 0.05 & 0.00 & 0.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/2$} \\ \hline
\textbf{50} &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Selection of $r$ based on Johansen Trace statistics
($p=0.05$)} \\ \hline
\textbf{50} &  & 0.28 & 0.55 & 0.69 &  & 0.41 & 0.29 & 0.22 &  & 0.14 & 0.06
& 0.03 &  & 0.17 & 0.10 & 0.06 \\
\textbf{500} &  & 0.28 & 0.54 & 0.69 &  & 0.41 & 0.29 & 0.22 &  & 0.14 & 0.06
& 0.03 &  & 0.17 & 0.10 & 0.06 \\
\textbf{1,000} &  & 0.28 & 0.54 & 0.69 &  & 0.41 & 0.29 & 0.22 &  & 0.14 &
0.06 & 0.03 &  & 0.17 & 0.10 & 0.06 \\
\textbf{3,000} &  & 0.28 & 0.54 & 0.69 &  & 0.41 & 0.29 & 0.22 &  & 0.14 &
0.06 & 0.03 &  & 0.17 & 0.10 & 0.06 \\ \hline
\multicolumn{9}{l}{\textbf{B. Experiments with }$r_{0}=1$} &  &  &  &  &  &
&  &  \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/4$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/2$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Selection of $r$ based on Johansen Trace statistics
($p=0.05$)} \\ \hline
\textbf{50} &  & 0.46 & 0.26 & 0.01 &  & 0.36 & 0.57 & 0.82 &  & 0.07 & 0.08
& 0.08 &  & 0.11 & 0.09 & 0.08 \\
\textbf{500} &  & 0.46 & 0.26 & 0.02 &  & 0.36 & 0.57 & 0.82 &  & 0.07 & 0.08
& 0.08 &  & 0.11 & 0.09 & 0.08 \\
\textbf{1,000} &  & 0.46 & 0.26 & 0.01 &  & 0.36 & 0.57 & 0.82 &  & 0.07 &
0.08 & 0.08 &  & 0.11 & 0.09 & 0.08 \\
\textbf{3,000} &  & 0.46 & 0.26 & 0.01 &  & 0.36 & 0.57 & 0.82 &  & 0.07 &
0.08 & 0.08 &  & 0.11 & 0.09 & 0.08 \\ \hline
\multicolumn{9}{l}{\textbf{C. Experiments with }$r_{0}=2$} &  &  &  &  &  &
&  &  \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/4$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/2$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Selection of $r$ based on Johansen Trace statistics
($p=0.05$)} \\ \hline
\textbf{50} &  & 0.36 & 0.06 & 0.00 &  & 0.39 & 0.61 & 0.43 &  & 0.09 & 0.15
& 0.31 &  & 0.16 & 0.18 & 0.25 \\
\textbf{500} &  & 0.36 & 0.06 & 0.00 &  & 0.39 & 0.61 & 0.43 &  & 0.09 & 0.15
& 0.31 &  & 0.16 & 0.18 & 0.25 \\
\textbf{1,000} &  & 0.36 & 0.06 & 0.00 &  & 0.39 & 0.61 & 0.43 &  & 0.09 &
0.15 & 0.31 &  & 0.16 & 0.18 & 0.25 \\
\textbf{3,000} &  & 0.36 & 0.06 & 0.00 &  & 0.39 & 0.61 & 0.43 &  & 0.09 &
0.15 & 0.31 &  & 0.16 & 0.18 & 0.25 \\ \hline\hline
\end{tabular}
\vspace*{-0.45in}
\end{center}

\begin{flushleft}
\singlespacing
\scriptsize
\singlespacing
Notes: For $r_{0}=0,$ (panel A) there are 3 experiments: with low, medium
and high serial correlation in first-difference VAR(1)\ model. For $r_{0}=1,$
(panel B) and $r_{0}=2$ (panel C) there are 8 VAR(1) experiments differing
in terms of the error distributions (Gaussian or chi-square), fit (high or
low), and speed of convergence towards long run (moderate or low). Lag order
for the computation of Johansen's trace statistics is set equal to the true
lag order for the VAR. All experiments based on $R=2,000$ MC replications.
Results for individual Monte Carlo experiments are available from the
authors upon request. A summary of the different MC designs is given in
Subsection \ref{DGPsubsection}, with a detailed account of the data
generating processes provided in Section \ref{Sup_MCd} of the
supplement.\pagebreak
\end{flushleft}

\begin{center}
\normalsize
\singlespacing
TABLE 2: Selection frequencies, averaged across experiments, for the
estimation of $r_{0}=0,1,2,3$ by eigenvalue thresholding estimator with $
{\Greekmath 010E} =1/4$ and $1/2$ and by Johansen's trace statistics using VARMA(1,1)
as the DGP with $r_{0}=1,2,$ and without interactive time effects.\smallskip

\setlength{\tabcolsep}{4pt}
\scriptsize
\begin{tabular}{rrrrrrrrrrrrrrrrr}
\hline\hline
&  & \multicolumn{3}{c}{Frequency $\tilde{r}=0$} &  & \multicolumn{3}{c}{
Frequency $\tilde{r}=1$} &  & \multicolumn{3}{c}{Frequency $\tilde{r}=2$} &
& \multicolumn{3}{c}{Frequency $\tilde{r}=3$} \\
\cline{3-5}\cline{7-9}\cline{11-13}\cline{15-17}
$n$ $\backslash $ $T$ &  & \textbf{20} & \textbf{50} & \textbf{100} &  &
\textbf{20} & \textbf{50} & \textbf{100} &  & \textbf{20} & \textbf{50} &
\textbf{100} &  & \textbf{20} & \textbf{50} & \textbf{100} \\ \hline
\multicolumn{9}{l}{\textbf{A. Experiments with }$r_{0}=1$} &  &  &  &  &  &
&  &  \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/4$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/2$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 & 0.00
& 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00 & 1.00 &  & 0.00 &
0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Selection of $r$ based on Johansen Trace statistics
($p=0.05$)} \\ \hline
\textbf{50} &  & 0.40 & 0.33 & 0.14 &  & 0.38 & 0.48 & 0.69 &  & 0.09 & 0.09
& 0.09 &  & 0.13 & 0.11 & 0.09 \\
\textbf{500} &  & 0.40 & 0.33 & 0.14 &  & 0.38 & 0.48 & 0.69 &  & 0.09 & 0.09
& 0.09 &  & 0.13 & 0.11 & 0.09 \\
\textbf{1,000} &  & 0.40 & 0.33 & 0.14 &  & 0.38 & 0.48 & 0.69 &  & 0.09 &
0.09 & 0.09 &  & 0.13 & 0.11 & 0.09 \\
\textbf{3,000} &  & 0.40 & 0.33 & 0.14 &  & 0.38 & 0.48 & 0.69 &  & 0.09 &
0.09 & 0.09 &  & 0.13 & 0.11 & 0.09 \\ \hline
\multicolumn{9}{l}{\textbf{B. Experiments with }$r_{0}=2$} &  &  &  &  &  &
&  &  \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/4$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Correlation matrix eigenvalue thresholding
estimator $\tilde{r}$, with ${\Greekmath 010E} =1/2$} \\ \hline
\textbf{50} &  & 0.00 & 0.00 & 0.00 &  & 0.02 & 0.00 & 0.00 &  & 0.98 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{500} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 & 1.00
& 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{1,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\
\textbf{3,000} &  & 0.00 & 0.00 & 0.00 &  & 0.00 & 0.00 & 0.00 &  & 1.00 &
1.00 & 1.00 &  & 0.00 & 0.00 & 0.00 \\ \hline
&  & \multicolumn{15}{l}{Selection of $r$ based on Johansen Trace statistics
($p=0.05$)} \\ \hline
\textbf{50} &  & 0.42 & 0.34 & 0.10 &  & 0.37 & 0.42 & 0.47 &  & 0.08 & 0.09
& 0.21 &  & 0.13 & 0.15 & 0.21 \\
\textbf{500} &  & 0.42 & 0.34 & 0.10 &  & 0.37 & 0.42 & 0.47 &  & 0.08 & 0.10
& 0.21 &  & 0.13 & 0.15 & 0.22 \\
\textbf{1,000} &  & 0.42 & 0.34 & 0.10 &  & 0.37 & 0.42 & 0.47 &  & 0.08 &
0.09 & 0.21 &  & 0.13 & 0.15 & 0.22 \\
\textbf{3,000} &  & 0.42 & 0.34 & 0.10 &  & 0.37 & 0.42 & 0.47 &  & 0.08 &
0.09 & 0.21 &  & 0.13 & 0.15 & 0.22 \\ \hline\hline
\end{tabular}
\vspace{-0.2in}
\end{center}

\begin{flushleft}
\singlespacing
\scriptsize
Notes: For $r_{0}=1,$ (panel A) and $r_{0}=2$ (panel B) there are 8
VARMA(1,1) experiments differing in terms of the error distributions
(Gaussian or chi-squared), fit (high or low), and speed of convergence
towards long run (moderate or low). Lag order for the computation of
Johansen's trace statistics is set equal to the integer part of $T^{-1/3}$.
All experiments are based on $R=2,000$ MC replications.\ Results for
individual Monte Carlo experiments are available from the authors upon
request. A summary of the different MC designs is given in Subsection \ref
{DGPsubsection}, with a detailed account of the data generating processes
provided in Section \ref{Sup_MCd} of the supplement.
\end{flushleft}

\normalsize
\onehalfspacing

\pagebreak

\begin{center}
\singlespacing
TABLE 3: Simulated bias, RMSE, size and power, averaged across 8
experiments, for PME and MG-Johansen estimators of long-run relations, using
a three variable VAR(1) with $r_{0}=2$ and without interactive time
effects\smallskip

\setlength{\tabcolsep}{4pt}
\scriptsize
\begin{tabular}{rrrrrr|rrrr|rrrrrrrr}
\hline\hline
&  & \multicolumn{3}{c}{\textbf{Bias (}$\times 100$\textbf{)}} &  &
\multicolumn{3}{|c}{\textbf{RMSE (}$\times 100$\textbf{)}} &  &
\multicolumn{3}{|c}{\textbf{Size}($\times 100$)} & \multicolumn{1}{r|}{} &
\multicolumn{3}{|c}{\textbf{Power (}$\times 100$\textbf{)}} &  \\
\cline{2-18}
$n\backslash T$ &  & \textbf{20} & \textbf{50} & \textbf{100} &  & \textbf{20
} & \textbf{50} & \textbf{100} &  & \textbf{20} & \textbf{50} & \textbf{100}
&  & \multicolumn{1}{|r}{\textbf{20}} & \textbf{50} & \textbf{100} &  \\
\hline
\multicolumn{13}{l}{\textbf{A. Results for }${\Greekmath 010C} _{13,0}$} &
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} &
\multicolumn{1}{l}{} &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=2$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.15 & -0.13 & -0.07 &  & 4.06 & 1.96 & 1.10 &  & 7.66 &
7.10 & 7.39 &  & \multicolumn{1}{|r}{15.21} & 42.18 & 81.30 &  \\
\textbf{500} &  & -0.02 & -0.15 & -0.06 &  & 1.34 & 0.64 & 0.35 &  & 7.13 &
5.83 & 5.17 &  & \multicolumn{1}{|r}{67.73} & 99.63 & 100.00 &  \\
\textbf{1,000} &  & 0.01 & -0.15 & -0.07 &  & 0.97 & 0.46 & 0.25 &  & 7.64 &
6.73 & 6.48 &  & \multicolumn{1}{|r}{90.24} & 100.00 & 100.00 &  \\
\textbf{3,000} &  & -0.05 & -0.14 & -0.06 &  & 0.66 & 0.29 & 0.15 &  & 12.84
& 8.16 & 7.76 &  & \multicolumn{1}{|r}{99.96} & 100.00 & 100.00 &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=4$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.62 & -0.39 & -0.12 &  & 3.29 & 1.59 & 0.79 &  & 7.48 &
8.33 & 7.63 &  & \multicolumn{1}{|r}{23.13} & 64.41 & 96.21 &  \\
\textbf{500} &  & -0.49 & -0.39 & -0.11 &  & 1.22 & 0.62 & 0.27 &  & 10.58 &
13.04 & 7.58 &  & \multicolumn{1}{|r}{91.19} & 100.00 & 100.00 &  \\
\textbf{1,000} &  & -0.47 & -0.38 & -0.12 &  & 0.95 & 0.51 & 0.21 &  & 14.99
& 20.15 & 11.16 &  & \multicolumn{1}{|r}{99.31} & 100.00 & 100.00 &  \\
\textbf{3,000} &  & -0.53 & -0.38 & -0.12 &  & 0.76 & 0.43 & 0.15 &  & 35.96
& 47.89 & 20.91 &  & \multicolumn{1}{|r}{100.00} & 100.00 & 100.00 &  \\
\hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{
MG-Johansen estimator} &  \\ \hline
\textbf{50} &  & 15.78 & -3.14 & -0.39 &  & \TEXTsymbol{>}100 & \TEXTsymbol{>
}100 & 65.53 &  & 2.66 & 2.96 & 3.26 &  & \multicolumn{1}{|r}{2.67} & 10.45
& 56.48 &  \\
\textbf{500} &  & -12.34 & 1.92 & -0.86 &  & \TEXTsymbol{>}100 & \TEXTsymbol{
>}100 & 85.82 &  & 1.98 & 1.89 & 2.66 &  & \multicolumn{1}{|r}{2.27} & 10.77
& 65.62 &  \\
\textbf{1,000} &  & -4.83 & -1.07 & 0.18 &  & \TEXTsymbol{>}100 &
\TEXTsymbol{>}100 & 36.39 &  & 2.16 & 2.24 & 2.31 &  & \multicolumn{1}{|r}{
2.38} & 11.44 & 65.38 &  \\
\textbf{3,000} &  & -46.82 & -40.69 & -0.17 &  & \TEXTsymbol{>}100 &
\TEXTsymbol{>}100 & 65.14 &  & 2.21 & 2.01 & 2.09 &  & \multicolumn{1}{|r}{
2.34} & 12.24 & 65.83 &  \\ \hline
\multicolumn{13}{l}{\textbf{B. Results for }${\Greekmath 010C} _{23,0}$} &
\multicolumn{1}{l}{} & \multicolumn{1}{|l}{} & \multicolumn{1}{l}{} &
\multicolumn{1}{l}{} &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=2$ sub-samples} &  \\ \hline
\textbf{50} &  & 0.02 & -0.15 & -0.07 &  & 4.11 & 1.96 & 1.09 &  & 8.26 &
7.43 & 7.31 &  & \multicolumn{1}{|r}{15.78} & 41.90 & 81.03 &  \\
\textbf{500} &  & -0.21 & -0.16 & -0.07 &  & 1.34 & 0.65 & 0.35 &  & 6.99 &
5.88 & 5.63 &  & \multicolumn{1}{|r}{67.41} & 99.65 & 100.00 &  \\
\textbf{1,000} &  & -0.07 & -0.14 & -0.07 &  & 0.98 & 0.46 & 0.25 &  & 7.53
& 6.32 & 6.61 &  & \multicolumn{1}{|r}{90.25} & 100.00 & 100.00 &  \\
\textbf{3,000} &  & -0.08 & -0.14 & -0.07 &  & 0.66 & 0.29 & 0.16 &  & 13.13
& 8.56 & 7.98 &  & \multicolumn{1}{|r}{99.97} & 100.00 & 100.00 &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=4$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.42 & -0.38 & -0.12 &  & 3.30 & 1.57 & 0.77 &  & 7.89 &
8.05 & 6.72 &  & \multicolumn{1}{|r}{23.71} & 64.31 & 96.66 &  \\
\textbf{500} &  & -0.71 & -0.40 & -0.12 &  & 1.31 & 0.63 & 0.28 &  & 13.23 &
13.69 & 8.26 &  & \multicolumn{1}{|r}{91.11} & 100.00 & 100.00 &  \\
\textbf{1,000} &  & -0.56 & -0.38 & -0.12 &  & 0.99 & 0.51 & 0.21 &  & 16.85
& 19.71 & 10.71 &  & \multicolumn{1}{|r}{99.25} & 100.00 & 100.00 &  \\
\textbf{3,000} &  & -0.57 & -0.38 & -0.12 &  & 0.78 & 0.43 & 0.16 &  & 37.17
& 48.16 & 21.74 &  & \multicolumn{1}{|r}{100.00} & 100.00 & 100.00 &  \\
\hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{
MG-Johansen estimator} &  \\ \hline
\textbf{50} &  & -21.74 & 3.58 & 0.52 &  & \TEXTsymbol{>}100 & \TEXTsymbol{>}
100 & 65.06 &  & 2.56 & 2.74 & 3.43 &  & \multicolumn{1}{|r}{14.82} & 15.76
& 56.87 &  \\
\textbf{500} &  & -6.62 & -1.29 & 0.65 &  & \TEXTsymbol{>}100 & \TEXTsymbol{>
}100 & 79.29 &  & 2.23 & 1.94 & 2.58 &  & \multicolumn{1}{|r}{15.19} & 15.74
& 65.54 &  \\
\textbf{1,000} &  & 13.31 & 0.70 & -0.18 &  & \TEXTsymbol{>}100 &
\TEXTsymbol{>}100 & 40.82 &  & 2.34 & 2.14 & 2.56 &  & \multicolumn{1}{|r}{
15.74} & 15.99 & 65.71 &  \\
\textbf{3,000} &  & 13.99 & 55.84 & 0.04 &  & \TEXTsymbol{>}100 &
\TEXTsymbol{>}100 & 56.04 &  & 2.09 & 2.17 & 2.23 &  & \multicolumn{1}{|r}{
15.49} & 17.06 & 65.86 &  \\ \hline
\end{tabular}
\vspace{-0.2in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: The long-run relations are given by $\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }
\mathbf{w}_{it}=w_{it,1}-w_{it,3}$ and $\mathbf{{\Greekmath 010C} }_{2,0}^{\prime }
\mathbf{w}_{it}=w_{it,2}-w_{it,3}$, and identified using ${\Greekmath 010C}
_{11,0}={\Greekmath 010C} _{22,0}=1$, and ${\Greekmath 010C} _{12,0}={\Greekmath 010C} _{21,0}=0$. The 8
experiments differ with respect to distribution (Gaussian or chi-squared),
fit (high or low), and speed of convergence toward long run (moderate or
low). Size and power are computed at the five percent nominal level.
Reported results are based on $R=2,000$ Monte Carlo replications. Simulated
power are computed under $H_{1}:$ ${\Greekmath 010C} _{13}=-0.97$, ${\Greekmath 010C} _{23}=-0.97$,
as alternatives to $-1$ for both coefficients under the null. Results for
individual Monte Carlo experiments are available from the authors upon
request. A summary of the different MC designs is given in Subsection \ref
{DGPsubsection}, with a detailed account of the data generating processes
provided in Section \ref{Sup_MCd} of the supplement.
\end{flushleft}

\normalsize
\onehalfspacing

\subsection{Estimation of coefficients of long-run relations\label
{coefficients}}

\subsubsection{Comparison of PME with MG-Johansen in VAR(1) designs with
multiple long-run relations}

We first investigate how PME estimators of multiple long-run relations
compare with the Mean Group estimator based on Johansen's estimator of
individual cointegrating vectors (MG-Johansen). To this end, we focus on the
VAR(1) design with multiple long-run relations ($r_{0}=2$ long-run relations
among $m=3$ variables) and no interactive time effects. We assume the true
lag order is known in the case of MG-Johansen. This is a set up which we
believe is most favorable to Johansen procedure when applied to the
individual units. But we note that the PME estimator does not require the
knowledge of the true lag order. These long-run relations are identified
according to (\ref{r2a})-(\ref{r2b}) for both PME, and MG-Johansen's
estimators. While moments of Johansen's estimator do not exist, we expect
the simulated size and power of MG-Johansen to be good for a sufficiently
large $T$ relative to $n$.

Table 3 provides a summary of the results for estimation of ${\Greekmath 010C} _{13,0}$
(\thinspace $=$ $-1)$ in part A, and for ${\Greekmath 010C} _{23,0}$ ($=-1$) in part B.
The table report bias, root mean square error (RMSE), size of the tests at
5\% nominal level, and power of the tests ($H_{1}:$ ${\Greekmath 010C} _{13}=-0.97$, $
{\Greekmath 010C} _{23,0}=-0.97$). All entries in this table are multiplied by $100$,
and averaged across $8$ experiments, defined in terms of error distributions
(Gaussian or chi-squared), fit (high or low), and speed of convergence
towards the long run (moderate or low).

We first note that the PME estimator works quite well, with relatively small
bias and RMSE that decline with $n$ and $T$. The choice of $q=2$ works well
for most $n$ and $T$ combinations, and $q=4$ performs better in terms of
RMSE only when $T=100$. In terms of size, PME with $q=2$ does better than
with $q=4$ for all $n$ and $T$ combinations, and has size close to the
nominal $5$ per cent level, together with satisfactory power that rapidly
rises to $100$ per cent as $n$ and $T$ are increased. However, even when $
q=2 $ there is some evidence of moderate size distortions, particularly when
$T=20$ and $n=3,000$. The size distortion of PME when $q=4$ seems to be
largely due to its bias that does not decline with $T$, despite the fall we
observe in its RMSE.

The results for the MG-Johansen estimator, also reported in Table 3, show
size below 5 percent and a very low power in comparison to PME. The very
large RMSE and bias entries could be due to lack of moments for the
unit-specific estimators obtained using the Johansen procedure. Clearly,
further research is needed for adapting the use of Johansen maximum
likelihood approach for use with large panels. We are using MG-Johansen
estimators here only for the purpose of comparisons when $r_{0}>1$. Below we
do consider other panel cointegration procedures when $r_{0}=1$, and we will
no longer report results for MG-Johansen as they are very similar to those
summarized in Table 3.

The above summary results do not depend much on which of the two
coefficients is considered.\pagebreak

\begin{center}
\singlespacing
TABLE 4: Simulated bias, RMSE, size and power, averaged across 8
experiments, for PME estimators of long-run relations in the case of a three
variables VARMA(1,1) with $r_{0}=2$, and without interactive time
effects\smallskip

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rrrrrr|rrrr|rrrr|rrrr}
\hline\hline
&  & \multicolumn{3}{c}{\textbf{Bias (}$\times 100$\textbf{)}} &  &
\multicolumn{3}{|c}{\textbf{RMSE (}$\times 100$\textbf{)}} &  &
\multicolumn{3}{|c}{\textbf{Size}($\times 100$)} &  & \multicolumn{3}{|c}{
\textbf{Power (}$\times 100$\textbf{)}} &  \\ \cline{2-18}
$n\backslash T$ &  & \textbf{20} & \textbf{50} & \textbf{100} &  & \textbf{20
} & \textbf{50} & \textbf{100} &  & \textbf{20} & \textbf{50} & \textbf{100}
&  & \textbf{20} & \textbf{50} & \textbf{100} &  \\ \hline
\multicolumn{15}{l}{\textbf{A. Results for }${\Greekmath 010C} _{13,0}$} &
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=2$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.20 & -0.27 & -0.14 &  & 6.92 & 4.05 & 2.44 &  & 8.49 &
7.43 & 7.99 &  & 10.44 & 16.84 & 33.66 &  \\
\textbf{500} &  & 0.16 & -0.22 & -0.13 &  & 2.24 & 1.31 & 0.77 &  & 7.25 &
5.80 & 5.81 &  & 29.62 & 72.41 & 96.88 &  \\
\textbf{1,000} &  & 0.23 & -0.20 & -0.13 &  & 1.66 & 0.93 & 0.55 &  & 8.40 &
5.98 & 5.79 &  & 48.41 & 92.58 & 99.92 &  \\
\textbf{3,000} &  & 0.16 & -0.21 & -0.11 &  & 1.08 & 0.56 & 0.32 &  & 12.96
& 6.78 & 6.73 &  & 86.03 & 99.99 & 100.00 &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=4$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.83 & -0.73 & -0.24 &  & 5.60 & 3.30 & 1.80 &  & 7.13 &
7.39 & 7.52 &  & 12.72 & 25.58 & 52.34 &  \\
\textbf{500} &  & -0.51 & -0.64 & -0.20 &  & 1.94 & 1.20 & 0.60 &  & 8.33 &
9.94 & 7.17 &  & 53.05 & 92.56 & 99.86 &  \\
\textbf{1,000} &  & -0.46 & -0.63 & -0.21 &  & 1.50 & 0.96 & 0.44 &  & 10.73
& 13.52 & 8.18 &  & 76.78 & 99.49 & 100.00 &  \\
\textbf{3,000} &  & -0.53 & -0.63 & -0.20 &  & 1.11 & 0.75 & 0.30 &  & 24.57
& 32.18 & 14.53 &  & 98.21 & 100.00 & 100.00 &  \\ \hline
\multicolumn{15}{l}{\textbf{B. Results for }${\Greekmath 010C} _{23,0}$} &
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=2$ sub-samples} &  \\ \hline
\textbf{50} &  & 0.06 & -0.23 & -0.12 &  & 6.84 & 4.07 & 2.40 &  & 8.50 &
7.71 & 7.47 &  & 10.87 & 17.45 & 33.51 &  \\
\textbf{500} &  & -0.06 & -0.25 & -0.13 &  & 2.25 & 1.31 & 0.77 &  & 6.94 &
5.81 & 6.01 &  & 29.10 & 72.41 & 96.82 &  \\
\textbf{1,000} &  & 0.11 & -0.21 & -0.13 &  & 1.64 & 0.93 & 0.55 &  & 8.03 &
5.81 & 5.99 &  & 48.22 & 92.34 & 99.93 &  \\
\textbf{3,000} &  & 0.12 & -0.20 & -0.11 &  & 1.09 & 0.56 & 0.33 &  & 12.66
& 6.92 & 7.06 &  & 86.06 & 99.98 & 100.00 &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=4$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.51 & -0.65 & -0.22 &  & 5.48 & 3.28 & 1.74 &  & 7.36 &
7.20 & 6.68 &  & 13.36 & 26.04 & 51.75 &  \\
\textbf{500} &  & -0.77 & -0.67 & -0.21 &  & 2.04 & 1.21 & 0.60 &  & 9.54 &
10.01 & 7.12 &  & 52.29 & 92.70 & 99.88 &  \\
\textbf{1,000} &  & -0.59 & -0.63 & -0.21 &  & 1.53 & 0.96 & 0.44 &  & 11.66
& 14.24 & 8.08 &  & 76.76 & 99.49 & 100.00 &  \\
\textbf{3,000} &  & -0.57 & -0.63 & -0.20 &  & 1.13 & 0.75 & 0.31 &  & 24.94
& 32.18 & 14.91 &  & 98.21 & 100.00 & 100.00 &  \\ \hline\hline
\end{tabular}
\vspace{-0.2in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: The long-run relations are given by $\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }
\mathbf{w}_{it}=w_{it,1}-w_{it,3}$ and $\mathbf{{\Greekmath 010C} }_{2,0}^{\prime }
\mathbf{w}_{it}=w_{it,2}-w_{it,3}$, and identified using ${\Greekmath 010C}
_{11,0}={\Greekmath 010C} _{22,0}=1$, and ${\Greekmath 010C} _{12,0}={\Greekmath 010C} _{21,0}=0$. The 8
experiments differ with respect to distribution (Gaussian or chi-squared),
fit (high or low), and speed of convergence toward long run (moderate or
low). Reported results are based on $R=2,000$ Monte Carlo replications.
Simulated power are computed under $H_{1}:$ ${\Greekmath 010C} _{13}=-0.97$ and ${\Greekmath 010C}
_{23}=-0.97$, as alternatives to $-1$ for both coefficients under the null.
Results for individual Monte Carlo experiments are available from the
authors upon request. A summary of the different MC designs is given in
Subsection \ref{DGPsubsection}, with a detailed account of the data
generating processes provided in Section \ref{Sup_MCd} of the supplement.
Size and Power are computed at 5 percent nominal level. \vspace{-0.3in}
\end{flushleft}

\begin{center}
\normalsize
\singlespacing
TABLE 5: Simulated bias, RMSE, size and power, averaged across 8
experiments, for PME estimators of long-run relations in the case of three
variables VARMA(1,1) with $r_{0}=2$, and with interactive time
effects\smallskip

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rrrrrr|rrrr|rrrr|rrrr}
\hline\hline
&  & \multicolumn{3}{c}{\textbf{Bias (}$\times 100$\textbf{)}} &  &
\multicolumn{3}{|c}{\textbf{RMSE (}$\times 100$\textbf{)}} &  &
\multicolumn{3}{|c}{\textbf{Size}($\times 100$)} &  & \multicolumn{3}{|c}{
\textbf{Power (}$\times 100$\textbf{)}} &  \\ \cline{2-18}
$n\backslash T$ &  & \textbf{20} & \textbf{50} & \textbf{100} &  & \textbf{20
} & \textbf{50} & \textbf{100} &  & \textbf{20} & \textbf{50} & \textbf{100}
&  & \textbf{20} & \textbf{50} & \textbf{100} &  \\ \hline
\multicolumn{15}{l}{\textbf{A. Results for }${\Greekmath 010C} _{13,0}$} &
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=2$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.20 & -0.26 & -0.14 &  & 6.89 & 4.05 & 2.44 &  & 8.56 &
7.46 & 8.00 &  & 10.59 & 16.85 & 33.76 &  \\
\textbf{500} &  & 0.16 & -0.21 & -0.13 &  & 2.23 & 1.31 & 0.77 &  & 7.15 &
5.83 & 5.79 &  & 29.86 & 72.38 & 96.91 &  \\
\textbf{1,000} &  & 0.24 & -0.20 & -0.13 &  & 1.65 & 0.93 & 0.55 &  & 8.38 &
5.94 & 5.76 &  & 48.74 & 92.63 & 99.92 &  \\
\textbf{3,000} &  & 0.16 & -0.21 & -0.11 &  & 1.08 & 0.56 & 0.32 &  & 12.91
& 6.78 & 6.71 &  & 86.23 & 99.99 & 100.00 &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=4$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.81 & -0.72 & -0.23 &  & 5.54 & 3.29 & 1.80 &  & 7.04 &
7.41 & 7.53 &  & 12.85 & 25.58 & 52.38 &  \\
\textbf{500} &  & -0.50 & -0.63 & -0.20 &  & 1.92 & 1.20 & 0.60 &  & 8.21 &
9.96 & 7.13 &  & 53.69 & 92.61 & 99.85 &  \\
\textbf{1,000} &  & -0.44 & -0.63 & -0.20 &  & 1.47 & 0.95 & 0.44 &  & 10.59
& 13.56 & 8.22 &  & 77.28 & 99.50 & 100.00 &  \\
\textbf{3,000} &  & -0.52 & -0.63 & -0.20 &  & 1.09 & 0.75 & 0.30 &  & 24.09
& 32.06 & 14.57 &  & 98.36 & 100.00 & 100.00 &  \\ \hline
\multicolumn{15}{l}{\textbf{B. Results for }${\Greekmath 010C} _{23,0}$} &
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=2$ sub-samples} &  \\ \hline
\textbf{50} &  & 0.05 & -0.22 & -0.12 &  & 6.81 & 4.06 & 2.40 &  & 8.41 &
7.76 & 7.54 &  & 11.00 & 17.61 & 33.53 &  \\
\textbf{500} &  & -0.06 & -0.24 & -0.13 &  & 2.23 & 1.31 & 0.77 &  & 6.90 &
5.79 & 6.00 &  & 29.31 & 72.36 & 96.86 &  \\
\textbf{1,000} &  & 0.11 & -0.21 & -0.13 &  & 1.64 & 0.93 & 0.55 &  & 7.99 &
5.81 & 6.06 &  & 48.69 & 92.39 & 99.93 &  \\
\textbf{3,000} &  & 0.12 & -0.20 & -0.11 &  & 1.08 & 0.56 & 0.33 &  & 12.46
& 6.91 & 7.06 &  & 86.13 & 99.98 & 100.00 &  \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{15}{l}{PME
estimator with $q=4$ sub-samples} &  \\ \hline
\textbf{50} &  & -0.49 & -0.64 & -0.22 &  & 5.43 & 3.27 & 1.74 &  & 7.45 &
7.23 & 6.63 &  & 13.33 & 25.83 & 51.71 &  \\
\textbf{500} &  & -0.75 & -0.67 & -0.21 &  & 2.01 & 1.21 & 0.60 &  & 9.43 &
9.98 & 7.16 &  & 52.87 & 92.71 & 99.88 &  \\
\textbf{1,000} &  & -0.57 & -0.63 & -0.21 &  & 1.51 & 0.96 & 0.44 &  & 11.54
& 14.21 & 8.06 &  & 77.26 & 99.50 & 100.00 &  \\
\textbf{3,000} &  & -0.55 & -0.62 & -0.20 &  & 1.11 & 0.75 & 0.31 &  & 24.68
& 32.24 & 14.89 &  & 98.32 & 100.00 & 100.00 &  \\ \hline
\end{tabular}
\vspace{-0.2in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: See the notes to Table 4.\pagebreak
\end{flushleft}

\begin{center}
\begin{tabular}{cc}
\multicolumn{2}{c}{Long run parameters} \\ \hline\hline
-${\Greekmath 010C} _{13,0}$ & -${\Greekmath 010C} _{23,0}$ \\
\multicolumn{1}{l}{A. $T=20$} & \multicolumn{1}{l}{} \\
\multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig1A1.jpg}} & \multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig1A2.jpg}} \\
\multicolumn{1}{l}{B. $T=50$} & \multicolumn{1}{l}{} \\
\multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig1B1.jpg}} & \multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig1B2.jpg}} \\
\multicolumn{1}{l}{C. $T=100$} & \multicolumn{1}{l}{} \\
\multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig1C1.jpg}} & \multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig1C2.jpg}}
\end{tabular}
\bigskip \bigskip \bigskip

FIGURE 1:\ Empirical power curves for the tests based on PME\ estimators of $
{\Greekmath 010C} _{13,0}$ and ${\Greekmath 010C} _{23,0}$ parameters of $r_{0}=2$ long-run
relations, using VARMA(1,1) without interactive time effects, slow speed of
convergence, $PR_{nT}^{2}=0.2$, and Gaussian errors. PME estimators use $q=2$
sub-samples. \pagebreak

\begin{tabular}{cc}
\multicolumn{2}{c}{Long run parameters} \\ \hline\hline
-${\Greekmath 010C} _{13,0}$ & -${\Greekmath 010C} _{23,0}$ \\
\multicolumn{1}{l}{A. $T=20$} & \multicolumn{1}{l}{} \\
\multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig2A1.jpg}} & \multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig2A2.jpg}} \\
\multicolumn{1}{l}{B. $T=50$} & \multicolumn{1}{l}{} \\
\multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig2B1.jpg}} & \multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig2B2.jpg}} \\
\multicolumn{1}{l}{C. $T=100$} & \multicolumn{1}{l}{} \\
\multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig2C1.jpg}} & \multicolumn{1}{l}{\includegraphics[width=0.3\textwidth]{Fig2C2.jpg}}
\end{tabular}
\bigskip \bigskip \bigskip

FIGURE 2:\ Empirical power curves for the tests based on PME\ estimators of $
{\Greekmath 010C} _{13,0}$ and ${\Greekmath 010C} _{23,0}$ parameters of $r_{0}=2$ long-run
relations, using VARMA(1,1) with interactive time effects, slow speed of
convergence, $PR_{nT}^{2}=0.2$, and Gaussian errors. PME estimators use $q=2$
sub-samples. \pagebreak

\singlespacing
TABLE 6: Simulated bias, RMSE, size and power for estimation of long-run
relation using VAR(1) with $m=2$ variables, $r_{0}=1$, and one-way long-run
causality\smallskip

\setlength{\tabcolsep}{4pt}
\scriptsize
\begin{tabular}{rrrrr|rrrr|rrrr|rrrr}
\hline\hline
& \multicolumn{4}{c|}{\textbf{Bias (}$\times 100$\textbf{)}} &
\multicolumn{4}{|c|}{\textbf{RMSE (}$\times 100$\textbf{)}} &
\multicolumn{4}{|c|}{\textbf{Size }($\times 100$)} & \multicolumn{4}{|c}{
\textbf{Power (}$\times 100$\textbf{)}} \\ \cline{2-17}
$n\backslash T$ & \textbf{20} & \textbf{50} & \textbf{100} &  & \textbf{20}
& \textbf{50} & \textbf{100} &  & \textbf{20} & \textbf{50} & \textbf{100} &
& \textbf{20} & \textbf{50} & \textbf{100} &  \\ \hline
& \multicolumn{16}{l}{PME estimator with $q=2$ sub-samples} \\ \hline
\textbf{50} & -0.98 & -0.19 & -0.03 &  & 4.92 & 2.11 & 1.05 &  & 7.95 & 6.90
& 6.30 &  & 15.90 & 37.65 & 82.40 &  \\
\textbf{500} & -0.98 & -0.16 & -0.06 &  & 1.77 & 0.70 & 0.34 &  & 10.35 &
6.75 & 5.70 &  & 78.30 & 99.85 & 100.00 &  \\
\textbf{1,000} & -0.93 & -0.17 & -0.06 &  & 1.42 & 0.51 & 0.24 &  & 15.90 &
7.40 & 4.70 &  & 96.30 & 100.00 & 100.00 &  \\
\textbf{3,000} & -0.91 & -0.18 & -0.05 &  & 1.09 & 0.32 & 0.14 &  & 33.00 &
10.50 & 5.60 &  & 100.00 & 100.00 & 100.00 &  \\ \hline
& \multicolumn{16}{l}{PME estimator with $q=4$ sub-samples} \\ \hline
\textbf{50} & -1.46 & -0.36 & -0.09 &  & 4.09 & 1.79 & 0.85 &  & 9.05 & 7.35
& 6.30 &  & 23.30 & 52.00 & 95.40 &  \\
\textbf{500} & -1.47 & -0.33 & -0.10 &  & 1.89 & 0.64 & 0.28 &  & 23.65 &
9.65 & 5.45 &  & 97.50 & 100.00 & 100.00 &  \\
\textbf{1,000} & -1.41 & -0.35 & -0.10 &  & 1.65 & 0.53 & 0.21 &  & 39.95 &
15.50 & 6.20 &  & 100.00 & 100.00 & 100.00 &  \\
\textbf{3,000} & -1.40 & -0.35 & -0.09 &  & 1.48 & 0.42 & 0.14 &  & 82.65 &
36.05 & 12.65 &  & 100.00 & 100.00 & 100.00 &  \\ \hline
& \multicolumn{16}{l}{System Pooled Mean Group Estimator} \\ \hline
\textbf{50} & -0.33 & 0.05 & -0.01 &  & 7.10 & 1.87 & 0.81 &  & 58.50 & 24.95
& 15.00 &  & 62.80 & 67.95 & 99.05 &  \\
\textbf{500} & -0.05 & 0.03 & 0.00 &  & 2.21 & 0.58 & 0.25 &  & 59.45 & 25.10
& 13.20 &  & 83.15 & 100.00 & 100.00 &  \\
\textbf{1,000} & -0.01 & 0.01 & 0.00 &  & 1.59 & 0.41 & 0.18 &  & 59.25 &
22.90 & 13.05 &  & 92.40 & 100.00 & 100.00 &  \\
\textbf{3,000} & -0.01 & 0.00 & 0.00 &  & 0.89 & 0.24 & 0.10 &  & 57.45 &
24.30 & 13.65 &  & 99.75 & 100.00 & 100.00 &  \\ \hline
& \multicolumn{16}{l}{Breitung's 2-Step Estimator} \\ \hline
\textbf{50} & 5.25 & 1.28 & 0.36 &  & 6.45 & 2.03 & 0.84 &  & 53.95 & 23.30
& 12.30 &  & 28.55 & 33.05 & 96.15 &  \\
\textbf{500} & 5.25 & 1.26 & 0.36 &  & 5.39 & 1.35 & 0.43 &  & 99.70 & 84.00
& 41.90 &  & 70.55 & 97.95 & 100.00 &  \\
\textbf{1,000} & 5.27 & 1.25 & 0.35 &  & 5.34 & 1.29 & 0.39 &  & 99.90 &
97.75 & 63.75 &  & 91.00 & 100.00 & 100.00 &  \\
\textbf{3,000} & 5.29 & 1.24 & 0.36 &  & 5.32 & 1.26 & 0.37 &  & 100.00 &
100.00 & 98.20 &  & 99.90 & 100.00 & 100.00 &  \\ \hline
& \multicolumn{16}{l}{Pooled Mean Group Estimator} \\ \hline
\textbf{50} & 1.69 & 0.40 & 0.08 &  & 5.63 & 1.76 & 0.78 &  & 44.90 & 19.90
& 10.85 &  & 43.55 & 59.50 & 98.70 &  \\
\textbf{500} & 1.91 & 0.36 & 0.08 &  & 2.52 & 0.64 & 0.25 &  & 66.25 & 28.45
& 12.80 &  & 52.45 & 100.00 & 100.00 &  \\
\textbf{1,000} & 1.92 & 0.33 & 0.08 &  & 2.25 & 0.50 & 0.19 &  & 79.65 &
33.55 & 14.35 &  & 61.00 & 100.00 & 100.00 &  \\
\textbf{3,000} & 1.90 & 0.32 & 0.08 &  & 2.02 & 0.39 & 0.13 &  & 98.30 &
56.95 & 22.00 &  & 80.95 & 100.00 & 100.00 &  \\ \hline
& \multicolumn{16}{l}{Pooled Bewley Estimator} \\ \hline
\textbf{50} & 3.70 & 0.75 & 0.19 &  & 5.17 & 1.70 & 0.76 &  & 20.05 & 10.20
& 7.20 &  & 7.15 & 35.65 & 96.80 &  \\
\textbf{500} & 3.76 & 0.73 & 0.18 &  & 3.92 & 0.86 & 0.29 &  & 92.40 & 34.15
& 12.65 &  & 11.60 & 99.95 & 100.00 &  \\
\textbf{1,000} & 3.78 & 0.71 & 0.18 &  & 3.87 & 0.79 & 0.24 &  & 99.80 &
58.55 & 19.80 &  & 18.45 & 100.00 & 100.00 &  \\
\textbf{3,000} & 3.80 & 0.71 & 0.18 &  & 3.82 & 0.73 & 0.21 &  & 100.00 &
95.75 & 50.20 &  & 41.95 & 100.00 & 100.00 &  \\ \hline
& \multicolumn{16}{l}{Panel FMOLS estimator} \\ \hline
\textbf{50} & 10.27 & 4.27 & 2.01 &  & 11.08 & 4.69 & 2.24 &  & 96.10 & 90.85
& 85.70 &  & 85.30 & 47.05 & 52.85 &  \\
\textbf{500} & 10.30 & 4.27 & 2.03 &  & 10.38 & 4.32 & 2.05 &  & 100.00 &
100.00 & 100.00 &  & 100.00 & 88.95 & 98.10 &  \\
\textbf{1,000} & 10.33 & 4.25 & 2.03 &  & 10.37 & 4.28 & 2.04 &  & 100.00 &
100.00 & 100.00 &  & 100.00 & 97.90 & 99.95 &  \\
\textbf{3,000} & 10.36 & 4.25 & 2.03 &  & 10.37 & 4.26 & 2.04 &  & 100.00 &
100.00 & 100.00 &  & 100.00 & 100.00 & 100.00 &  \\ \hline
& \multicolumn{16}{l}{Panel Dynamic OLS Estimator} \\ \hline
\textbf{50} & 4.15 & 1.15 & 0.36 &  & 6.51 & 2.15 & 0.91 &  & 23.45 & 15.10
& 9.40 &  & 12.85 & 24.85 & 90.25 &  \\
\textbf{500} & 4.13 & 1.12 & 0.35 &  & 4.44 & 1.25 & 0.43 &  & 72.85 & 58.95
& 30.50 &  & 16.05 & 94.50 & 100.00 &  \\
\textbf{1,000} & 4.19 & 1.11 & 0.35 &  & 4.35 & 1.18 & 0.39 &  & 87.45 &
84.20 & 50.80 &  & 21.65 & 99.80 & 100.00 &  \\
\textbf{3,000} & 4.21 & 1.10 & 0.35 &  & 4.26 & 1.13 & 0.37 &  & 91.25 &
99.45 & 93.10 &  & 39.50 & 99.80 & 100.00 &  \\ \hline\hline
\end{tabular}
\vspace{-0.2in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: The long-run relation is given by $\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }
\mathbf{w}_{it}=w_{it,1}-w_{it,2}={\Greekmath 010C} _{11,0}w_{it,1}+{\Greekmath 010C}
_{12,0}w_{it,2} $, and identified with ${\Greekmath 010C} _{11,0}=1$. Coefficient ${\Greekmath 010C}
_{12,0}=-1$ is estimated. Data generating process used for results reported
in this table is taken from
\citeN{ChudikPesaranSmith2021PB}
. Section 3.1 of
\citeN{ChudikPesaranSmith2021PB}
provides full account of this design. Reported results are based on $R=2,000$
Monte Carlo replications. Size and Power are computed at 5 percent nominal
level. Simulated powers are computed under $H_{1}:{\Greekmath 010C} _{12}=-0.97$,
compared to null value of $-1$. \pagebreak
\end{flushleft}

\normalsize
\onehalfspacing

\subsubsection{Performance of PME in VARMA(1,1) designs}

Results when using VARMA(1,1) designs with $r_{0}=2$ are presented in Tables
4 for models without time effects, and in Table 5 for models with
interactive time effects. As before, these results are averages across eight
VARMA(1,1) experiments featuring $r_{0}=2$ long-run relations that differ in
terms of error distributions (Gaussian or chi-squared), fit (high or low),
and speed of convergence toward long run (moderate or low).\ Figures 1-2
give empirical power curves for tests on ${\Greekmath 010C} _{13,0}$ and ${\Greekmath 010C} _{23,0}$
using PME estimators computed with $q=2$ sub-samples, in VARMA(1,1) designs
with $r_{0}=2$ long-run relations, and for $n=50,$ $500,$ and $1000.$
Separate panels show $T=20,$ $T=50,$ $T=100.$ This experiment has a slow
speed of convergence, $PR_{nT}^{2}=0.2,$ Gaussian errors and without
interactive time effects (Figure 1) or with interactive time effects (Figure
2). In both cases, the power increases with $n$ and $T$ as expected.

Qualitatively, the VARMA results in Table 4 are similar to the VAR results
summarized in Table 3, with one important exception. Allowing for an MA
component in the DGP tends to reduce power of the tests. For example, in the
case where $n=500$, $T=20$, and $q=2$, the power of testing ${\Greekmath 010C}
_{13,0}=-1 $ against the alternative ${\Greekmath 010C} _{13}=-0.97$ is $67.73$ per cent
when using VAR design (Table 3) as compared to $29.62$ per cent when using
the VARMA design (Table 4). Otherwise, the results for bias, RMSE and size
are comparable across the two designs. Similarly, allowing for interactive
time effects (Table 5) does not alter these conclusions, and PME seems to be
quite robust to the inclusion of interactive time effects, in line with the
theoretical results of Section \ref{IntEffects}, so long as the time effects
are not trended (deterministic or stochastic).

\subsubsection{Comparisons with available estimators in the design with
single long-run relation and one-way long-run causality\label{SubS_MC_c}}

Last but not least, we investigate how PME compares with existing approaches
for the estimation of a single long-run relation, in a design that is
favorable to single equation estimators, some of which rely on the direction
of long-run causality to be one-way and known, and the lag order of the VAR
to be well specified. We report results for panel FMOLS estimator by
\citeANP{Pedroni1996} (\citeyearNP{Pedroni1996}, \citeyearNP{Pedroni2001}, \citeyearNP{Pedroni2001ReStat})
, the Pooled Mean Group (PMG) estimator by
\citeN{PesaranShinSmith1999}
, panel Dynamic OLS (PDOLS) by
\citeN{MarkSul2003}
, the two-step system estimator of
\citeN{Breitung2005}
, the system PMG estimator of
\citeN{ChudikPesaranSmith2023}
, and the pooled Bewley estimator by
\citeN{ChudikPesaranSmith2021PB}
.\ For large $T$ panels with moderate $n$, we would expect these single
equation techniques to perform better than the PME that allows for MA
components and does not assume long-run causality. This is confirmed by the
results summarized in Table 6. When $T=100$ and $n=50$, PME ($q=2$) has
higher RMSE than all other estimators reported in Table 6 except for the
FMOLS estimator. In contrast, PME estimator ($q=2$) is much more balanced in
terms of bias, RMSE and size of the tests, when $T$ is small ($=20$), and $n$
quite large ($=1000$). None of the alternative estimators to PME in the case
of $r_{0}=1$ (and known one-way long-run causality) work well in samples
where $T$ is not very large and $n$ much larger than $T$.

In terms of bias and size, the PME estimator with $q=2$ performs much better
than all the other single equation estimators under consideration, even if
we consider $T=100$ and $n=50$. Overall, Monte Carlo findings show that the
PME estimator with $q=2$ can have satisfactory performance for panels where $
n$ is quite large relative to $T$, in particular for panels with $n$ and $T$
combinations similar to the ones we consider in our empirical application
discussed below.

\section{Empirical Applications\label{EA}}

We provide two empirical applications to illustrate wide applicability of
the PME approach to both micro and macro panels.

\subsection{Estimation of long-run financial relations}

The fist application considers micro panels of firms where $n$ is quite
large relative to the available time dimension. We use logarithms of six key
financial variables from CRSP/Compustat, available from Wharton Research
Data Services. The six variables (measured in logarithms) are book value ($
BV_{it}$), market value ($MV_{it}$), short-term debt ($SD_{it}$), long-term
debt ($LD_{it}$), total assets ($TA_{it}$) and total debt outstanding ($
DO_{it}$). These are variables among which one would expect some key
relations. We consider them in three sets of two or three variables, using
two unbalanced panels both begin in 1950, the shorter one ends in 2010 and
the longer one in 2021. The maximum $T$ is $71$. We set a minimum $T$ of $20$
, and the average $T$ is around 30. The number of firms $n$ varies from
about $1,000$ to $2,500$ depending on the set of variables under
consideration. As well as estimating the number of long-run relations and
their parameters we test whether the coefficients in the linear combinations
of logarithms take the value -1, to relate our results to the ratios used in
corporate finance. In corporate finance accounting ratios, constructed from
balance sheet data, are commonly used to measure the profitability,
liquidity, and solvency of a firm. The rationales for the use of ratios
include correcting for size in the cross section dimension and eliminating
common trends in the time series dimension to render the ratios stationary.
These two objectives are not always compatible. In an early contribution,
that remains relevant,
\citeN{LevSunder1979}
comment \textquotedblleft It appears that the extensive use of financial
ratios by both practitioners and researchers is often motivated by tradition
and convenience rather than resulting from theoretical considerations or
from a careful statistical analysis.\textquotedblright\
\shortciteN{Geelen_etal_2024}
provide a more recent study of the use of financial ratios in empirical
corporate finance literature.

Given that the theory is often not very specific, it is desirable to have a
statistical criteria to judge which are the appropriate long-run relations
among the set of variables considered and whether the logarithm of their
ratios is stationary. The time series stationarity of finance ratios like
the aggregate dividend price ratio have been studied, but the question has
not, to our knowledge, been addressed in corporate finance, where the
context is somewhat different. Corporate finance studies tend to use
unbalanced panels with large $n$ and relatively small $T$ and and the vector
of accounting variables, of the sort one gets from Compustat, may include
multiple long-run relations. Thus the PME estimator which is appropriate for
multiple long-run relations in a large $n$, moderate $T$ panel seems well
designed to determine whether there are long-run relations in accounting
data.

Using a multiplicative specification, the relation between $y_{it},$ $x_{it}$
and $z_{it}$ can be readily cast in terms of $\mathbf{w}_{it}=(\ln
y_{it},\ln x_{it},\ln z_{it})^{\prime }$ and the PME procedure can be used
to test (a) if $\mathbf{{\Greekmath 010C} }_{0}^{\prime }\mathbf{w}_{it}$ is stationary;
and (b) if ${\Greekmath 010C} _{11,0}=1,$ ${\Greekmath 010C} _{12,0}=-1,$ ${\Greekmath 010C} _{13,0}=0;$ to
validate the the use of $\ln \left( y_{it}/x_{it}\right) $ or $y_{it}/x_{it}$
in econometric analysis. If step (a) cannot be validated then $\mathbf{{\Greekmath 010C}
}_{0}^{\prime }\mathbf{w}_{it}$ will not be stationary and its use in
econometric analysis could lead to spurious results. But if step (a) is
validated but not (b), whilst it would not be advisable to use $\ln
y_{it}-\ln x_{it}$, one can still consider using $\mathbf{\tilde{{\Greekmath 010C}}}
^{\prime }\mathbf{w}_{it}$, where $\mathbf{\tilde{{\Greekmath 010C}}}$ could be the
exactly identified PME estimator of $\mathbf{{\Greekmath 010C} }_{0}$, in second stage
panel regressions that allow for short term dynamics as well as other
stationary variables.\footnote{
To simplify the notations, we use the symbol tilde in this section to denote
PME estimates of the exactly identified long-run relations.} Such a two-step
procedure is justified, noting that $\mathbf{\tilde{{\Greekmath 010C}}}$ is super
consistent, converging to $\mathbf{{\Greekmath 010C} }_{0}$ at the rate of $T\sqrt{n}$.

Data sources, full definitions of the variables, summary statistics and
additional details on construction of the samples for each of the three
variable sets are provided in Section \ref{Sup_micro} of the supplement. The
following data filters are applied sequentially to each variable set and
sample period, separately. Firms are omitted if, for a given variable set
and sample: they do not have data for all variables in the set (without gaps
and covering at least 20 time periods); have nonpositive entries on any of
the variables, since we are using logarithms; and average value of key
ratios fall below the 1st or above the 99th percentiles estimated after the
application of the first two filters. This is similar to the filtering
\shortciteN{Geelen_etal_2024}
use, except that we require a longer time series dimension.\footnote{
\shortciteN{Geelen_etal_2024}
state: \textquotedblleft We winsorize all variables at the 1\% and 99\%
levels to mitigate the impact of outliers. We drop all observations with
missing values on one or more variables of interest. We remove observations
with a market-to-book ratio larger than 20, negative book equity or negative
EBITDA. Our final sample consists of 68,833 firm-year observations with
6,001 unique firms.\textquotedblright\ This gives $\bar{T}=11.5,$ though
some regressions use less.}

The variables are grouped into three sets, where we have prior expectations
about possible cointegration amongst them. To illustrate the procedure we
start with the simplest case where $m=2$ and $\mathbf{w}_{it}$ include the
logarithm of total debt outstanding and logarithm of total assets: \{$
DO_{it} $, $TA_{it}$\}. The ratio of total debt to total assets is often
used as a measure of leverage, so there is a single hypothesized long-run
relation. To showcase the performance of the PME procedure with multiple
long-run relations, we consider two other sets of variables with $m=3$. They
are: the logarithms of short and long term debt and total assets, \{$SD_{it}$
, $LD_{it}$, $TA_{it}$\}; and the logarithms of total debt outstanding, book
value and market value, \{$DO_{it}$, $BV_{it}$, $MV_{it}$\}. We expect two
hypothesized long-run relations in both of these sets. Since these variables
are often used to construct accounting ratios, whether the long-run relation
has a unit coefficient is also of interest. Because of missing firm
observations on some variables, the number of firms with minimum of $20$
data points on all the variables under consideration falls as we include
more variables in a set. For this reason we do not consider all six
variables together.

Table 7\textit{\ }gives the estimates of the number of long-run relations
for two and three variable models for the 1950-2021 and 1950-2010 unbalanced
samples using the eigenvalue thresholding\textit{\ }procedure, given by (\ref
{r_est}). It also reports the eigenvalues of the correlation matrix, $
\mathbf{R}_{ww}$ defined by (\ref{Rww}), together with the associated
threshold values, $T_{ave}^{-{\Greekmath 010E} }$, where $T_{ave}=n^{-1}
\sum_{i=1}^{n}T_{i}$. Since the panel is unbalanced we base the thresholds
on $T_{ave}$ and provide $\tilde{r}$ for ${\Greekmath 010E} =1/2$ and ${\Greekmath 010E} =1/4$,
using $q=2$ sub-sample time averages.

The preferred threshold based on ${\Greekmath 010E} =1/4$ gives $\tilde{r}=1$ long-run
relation for the panel data models with $m=2,$ and $\tilde{r}=2$ long-run
relations for the two cases with $m=3$. The threshold with ${\Greekmath 010E} =1/2$
also yields $\tilde{r}=1$ in the case with $m=2$, but when used in the case
of panels with $m=3$ it selects $\tilde{r}=1$ rather than $\tilde{r}=2$.
Given the theoretical discussion above and the Monte Carlo results, we
proceed using the estimates of the number of long-run relations obtained
using the preferred value of ${\Greekmath 010E} =1/4$, namely $one$ long-run relation
when $m=2$ and $two$ long-run relations when $m=3$.\footnote{
Table S10 in the supplement reports IPS panel unit root tests by
\shortciteN{ImPesaranShin2003}
(using a 40-year balanced panel), which do not reject the null of unit root
in all cases except for short-term debt (SD). Given the $5$ per cent chance
that the IPS test could be in error, we proceed assuming that all the six
variables are $I(1)$.}

Tables 8-10 present PME estimates of the coefficients in the long-run
relations, their standard errors and $t$-statistics for testing the null
hypothesis that the long-run coefficient in question is equal to $-1$. The
first set of estimates, in Table 8, are for panels with $\mathbf{w}
_{it}=(DO_{it},TA_{it})^{\prime }$. Recall that $DO_{it}$ and $TA_{it}$ are
\textit{logarithms} of debt outstanding and total assets. There is a single
hypothesized long-run relation: ${\Greekmath 010C} _{11,0}DO_{it}+{\Greekmath 010C} _{12,0}TA_{it}$.
Panel A of Table 8 uses the exact identifying condition ${\Greekmath 010C} _{11,0}=1$
and provides PME estimates of ${\Greekmath 010C} _{12,0}$ and t-statistics for the null
value of ${\Greekmath 010C} _{12,0}=-1$. \ The PME estimates of ${\Greekmath 010C} _{12,0}$ at $
-1.142$ and $-1.113$ for the two sample periods ending in 2021 and 2010,
respectively, are similar and close to but significantly different from $-1$
. To illustrate that, unlike regression based methods, the PME estimator is
invariant to normalization, part B of Table 8 uses the exact identifying
condition ${\Greekmath 010C} _{12,0}=1$, and reports the PME estimate of ${\Greekmath 010C} _{11,0}$
. It is confirmed that up to rounding error $\tilde{{\Greekmath 010C}}_{11}=1/\tilde{
{\Greekmath 010C}}_{12}$.\pagebreak

\begin{center}
\singlespacing
TABLE 7:\ Estimates of the number of long-run relations ($\tilde{r}$) by
eigenvalue thresholding using firm-level data and $q=2$ sub-sample time
averages\smallskip

\setlength{\tabcolsep}{4pt}
\scriptsize
\begin{tabular}{rcccccrrrrrr}
\hline\hline
\multicolumn{2}{r}{Variables:} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{$
\mathbf{w}_{it}=\left( DO_{it},TA_{it}\right) ^{\prime }$} &  &
\multicolumn{2}{c}{$\mathbf{w}_{it}=\left( SD_{it},LD_{it},TA_{it}\right)
^{\prime }$} &  & \multicolumn{2}{c}{$\mathbf{w}_{it}=\left(
DO_{it},BV_{it},MV_{it}\right) ^{\prime }$} \\
\cline{3-4}\cline{5-6}\cline{8-9}\cline{11-12}
\multicolumn{2}{r}{Sample end year:} &  &  & {\footnotesize 1950-2021} &
{\footnotesize 1950-2010} &  & {\footnotesize 1950-2021} & {\footnotesize
1950-2010} &  & {\footnotesize 1950-2021} & {\footnotesize 1950-2010} \\
\cline{3-12}
\multicolumn{11}{l}{\textbf{Estimated number of long-run relations (}$\tilde{
r}$\textbf{)}} &  \\
$\tilde{r}$ (${\Greekmath 010E} =1/2$) &  &  &  & 1 & 1 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{1} & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{1} & \multicolumn{1}{c}{1} \\
$\tilde{r}$ (${\Greekmath 010E} =1/4$) &  &  &  & 1 & 1 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{2} & \multicolumn{1}{c}{2} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{2} & \multicolumn{1}{c}{2} \\ \hline
\multicolumn{3}{l}{\textbf{Eigenvalues}} &  &  &  & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \hline
$\tilde{{\Greekmath 0115}}_{1}$ &  &  &  & 0.090 & 0.079 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{0.106} & \multicolumn{1}{c}{0.100} & \multicolumn{1}{c}{}
& \multicolumn{1}{c}{0.058} & \multicolumn{1}{c}{0.046} \\
$\tilde{{\Greekmath 0115}}_{2}$ &  &  &  & 1.910 & 1.921 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{0.251} & \multicolumn{1}{c}{0.223} & \multicolumn{1}{c}{}
& \multicolumn{1}{c}{0.226} & \multicolumn{1}{c}{0.198} \\
$\tilde{{\Greekmath 0115}}_{3}$ &  &  &  & - & - & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{2.644} & \multicolumn{1}{c}{2.678} & \multicolumn{1}{c}{}
& \multicolumn{1}{c}{2.717} & \multicolumn{1}{c}{2.756} \\ \hline
\multicolumn{3}{l}{\textbf{Threshold }$T_{ave}^{-{\Greekmath 010E} }$} &
\multicolumn{1}{l}{} &  &  & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} \\ \hline
${\Greekmath 010E} =1/2$ &  &  &  & 0.176 & 0.178 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{0.177} & \multicolumn{1}{c}{0.179} & \multicolumn{1}{c}{}
& \multicolumn{1}{c}{0.178} & \multicolumn{1}{c}{0.182} \\
${\Greekmath 010E} =1/4$ &  &  &  & 0.419 & 0.422 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{0.421} & \multicolumn{1}{c}{0.423} & \multicolumn{1}{c}{}
& \multicolumn{1}{c}{0.422} & \multicolumn{1}{c}{0.426} \\ \hline
\multicolumn{3}{l}{\textbf{Sample dimensions}} & \multicolumn{1}{l}{} &  &
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \hline
$n$ &  &  &  & 2,555 & 1,901 & \multicolumn{1}{c}{} & \multicolumn{1}{c}{
1,373} & \multicolumn{1}{c}{1,101} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{1,415} & \multicolumn{1}{c}{1,164} \\
$T_{ave}$ &  &  &  & 32.4 & 31.6 & \multicolumn{1}{c}{} & \multicolumn{1}{c}{
31.8} & \multicolumn{1}{c}{31.1} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{
31.5} & \multicolumn{1}{c}{30.3} \\
$\max_{i}T_{i}$ &  &  &  & 73 & 61 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{73} & \multicolumn{1}{c}{61} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{61} & \multicolumn{1}{c}{49} \\
$\sum_{i=1}^{n}T_{i}$ &  &  &  & 82,837 & 60,118 & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{43,621} & \multicolumn{1}{c}{34,262} & \multicolumn{1}{c}{
} & \multicolumn{1}{c}{44,592} & \multicolumn{1}{c}{35,293} \\ \hline\hline
\end{tabular}
\vspace{-0.25in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: This table reports eigenvalues $\tilde{{\Greekmath 0115}}_{j}$ for $
j=1,2...,m\, $\ (in ascending order) of $\mathbf{R}_{_{\bar{w}\bar{w}}}$
given by (\ref{Rww}) using $q=2$ sub-sample time averages and the
corresponding eigenvalue thresholding estimates of the number of long-run
relations given by $\tilde{r}=\sum_{j=1}^{m}\mathcal{I}\left( \tilde{{\Greekmath 0115}}
_{j}<T_{ave}^{-{\Greekmath 010E} }\right) $, for ${\Greekmath 010E} =1/4,1/2,$ where $
T_{ave}=n^{-1}\sum_{i=1}^{n}T_{i}$, and $T_{i}\geq 20$ for all $i$. The
elements of $\mathbf{w}_{it}$ are logarithms of book value ($BV_{it}$),
market value ($MV_{it}$), short-term debt ($SD_{it}$), long-term debt ($
LD_{it}$), total assets ($TA_{it}$) and total debt outstanding ($DO_{it}$).
See Section \ref{Sup_micro} of the supplement for variable definitions, data
sources and availability, and filters applied.
\end{flushleft}

\normalsize
\onehalfspacing

For panels with $m=3$ and $\tilde{r}=2$ we need two exactly identifying
conditions on each of the two long-run relations. In the case where $\mathbf{
w}_{it}=(SD_{it},LD_{it},TA_{it})^{\prime }$, we use the conditions ${\Greekmath 010C}
_{11,0}=1$ and ${\Greekmath 010C} _{12,0}=0$ to exactly identify the first long-run
relation, and conditions ${\Greekmath 010C} _{21,0}=0$ and ${\Greekmath 010C} _{22,0}=1$ to identify
the second long-run relation. Hence the two identified long-run relations
are $SD_{it}+{\Greekmath 010C} _{13,0}TA_{it}$ and $LD_{it}+{\Greekmath 010C} _{23,0}TA_{it}$. PME
estimates for ${\Greekmath 010C} _{13,0}$ in Table 9 are -1.025 and -1.024 for the two
unbalanced samples, both very close to -1 and statistically not different
from -1. PME estimates for ${\Greekmath 010C} _{23,0}$ are -0.875 and -0.899, both
statistically different from -1 at the one percent level. Rotation of the
two identified long-run relations reported in Table 9 imply the long-run
relations $SD_{it}-1.156LD_{it}$ and $SD_{it}-1.130LD_{it}$ for the samples
ending in 2021 and 2010, respectively. Both of these PME estimates, $-1.156$
and $-1.130$, are statistically significantly different from $-1$ at the one
per cent level. Hence, in the case of the variable set $\left\{
SD_{it},LD_{it},TA_{it}\right\} $, the unit long-run elasticity hypothesis
could be accepted for short term debt to total assets only..\pagebreak

\begin{center}
\singlespacing
TABLE 8:\textit{\ }PME estimates for the set $\left\{
DO_{it},TA_{it}\right\} $, using $q=2$\ sub-sample time averages and one\
long-run relation

\setlength{\tabcolsep}{4pt}
\scriptsize
\begin{tabular}{rccc}
\multicolumn{4}{r}{\textbf{A. Exact identifying condition }${\Greekmath 010C} _{11,0}=1$}
\\ \hline\hline
\multicolumn{4}{r}{Exactly identified long-run relation} \\
& \multicolumn{3}{c}{$\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }\mathbf{w}
_{it}=DO_{it}+{\Greekmath 010C} _{12,0}TA_{it}$} \\ \hline
Sample ends: &  & {\footnotesize 1950-2021} & {\footnotesize 1950-2010} \\
\hline
\multicolumn{1}{l}{$\tilde{{\Greekmath 010C}}_{12}$} &  & -1.142 & -1.113 \\
\multicolumn{1}{l}{s.e.} &  & (0.010) & (0.011) \\
\multicolumn{1}{l}{t(${\Greekmath 010C} _{12,0}=-1)$} &  & -14.6 & -10.4 \\ \hline
\multicolumn{1}{l}{$n$} &  & 2,555 & 1,901 \\
\multicolumn{1}{l}{$\sum_{i=1}^{n}T_{i}$} &  & 82,837 & 60,118 \\
\hline\hline
\end{tabular}
\qquad \qquad\ \
\begin{tabular}{rccc}
\multicolumn{4}{r}{\textbf{B. Exact identifying condition }${\Greekmath 010C} _{12,0}=1$}
\\ \hline\hline
\multicolumn{4}{r}{Exactly identified long-run relation} \\
& \multicolumn{3}{c}{$\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }\mathbf{w}_{it}={\Greekmath 010C}
_{11,0}DO_{it}+TA_{it}$} \\ \hline
Sample ends: &  & {\footnotesize 1950-2021} & {\footnotesize 1950-2010} \\
\hline
\multicolumn{1}{l}{$\tilde{{\Greekmath 010C}}_{11}$} &  & -0.875 & -0.899 \\
\multicolumn{1}{l}{s.e.} &  & (0.010) & (0.012) \\
\multicolumn{1}{l}{t(${\Greekmath 010C} _{11,0}=-1)$} &  & -12.8 & -8.7 \\ \hline
\multicolumn{1}{l}{$n$} &  & 2,555 & 1,901 \\
\multicolumn{1}{l}{$\sum_{i=1}^{n}T_{i}$} &  & 82,837 & 60,118 \\
\hline\hline
\end{tabular}
\vspace{-0.2in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: $TA_{it}$ and $DO_{it}$ are logarithms of total assets and total debt
outstanding. Left panel reports PME estimate of ${\Greekmath 010C} _{12,0}$ using the
exact identifying condition ${\Greekmath 010C} _{11,0}=1$. Right panel reports PME
estimate of ${\Greekmath 010C} _{11,0}$ using the exact identifying condition ${\Greekmath 010C}
_{12,0}=1$. PME estimator, given by (\ref{BETAdot_hat}), is computed using $
q=2$\ sub-sample time averages, subject to $T_{i}\geq 20$, for all $i$. To
simplify the notations we use the tilde symbols to denote the exactly
identified PME estimates. The row labelled t(${\Greekmath 010C} _{12,0}=-1)$ and t($
{\Greekmath 010C} _{11,0}=-1)$ gives the t statistic for testing $H_{0}:{\Greekmath 010C} _{12,0}=-1$
and $H_{0}:{\Greekmath 010C} _{11,0}=-1$, respectively. See Section \ref{Sup_micro} of
the supplement for variable definitions, data sources and availability, and
filters applied.\vspace{-0.3in}
\end{flushleft}

\begin{center}
\singlespacing
TABLE 9:\textit{\ }PME estimates for the variable set $\left\{
SD_{it},LD_{it},TA_{it}\right\} $, using $q=2$\ sub-sample time averages and
two\ long-run relations

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rccccccc}
\hline\hline
& \multicolumn{2}{c}{First exactly identified} &  &  &  & \multicolumn{2}{c}{
Second exactly identified} \\
& \multicolumn{2}{c}{long-run relation} &  &  &  & \multicolumn{2}{c}{
long-run relation} \\
\multicolumn{1}{l}{} & \multicolumn{2}{c}{$\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }
\mathbf{w}_{it}=SD_{it}+{\Greekmath 010C} _{13,0}TA_{it}$} &  &  &  & \multicolumn{2}{c}{
$\mathbf{{\Greekmath 010C} }_{2,0}^{\prime }\mathbf{w}_{it}=LD_{it}+{\Greekmath 010C} _{23,0}TA_{it}$
} \\ \hline
Sample end year: & {\footnotesize 1950-2021} & {\footnotesize 1950-2010} &
&  & Sample end year: & {\footnotesize 1950-2021} & {\footnotesize 1950-2010}
\\ \hline
\multicolumn{1}{l}{$\tilde{{\Greekmath 010C}}_{13}$} & -1.025 & -1.024 &  &  &
\multicolumn{1}{l}{$\tilde{{\Greekmath 010C}}_{23}$} & -1.185 & -1.158 \\
\multicolumn{1}{l}{s.e.} & (0.019) & (0.019) &  &  & \multicolumn{1}{l}{s.e.}
& (0.015) & (0.016) \\
\multicolumn{1}{l}{t(${\Greekmath 010C} _{13,0}=-1)$} & -1.4 & -1.3 &  &  &
\multicolumn{1}{l}{t(${\Greekmath 010C} _{23,0}=-1)$} & -12.0 & -9.7 \\ \hline
\multicolumn{1}{l}{$n$} & 1,373 & 1,101 &  &  & \multicolumn{1}{l}{$n$} &
1,373 & 1,101 \\
\multicolumn{1}{l}{$\sum_{i=1}^{n}T_{i}$} & 43,621 & 34,262 &  &  &
\multicolumn{1}{l}{$\sum_{i=1}^{n}T_{i}$} & 43,621 & 34,262 \\ \hline\hline
\end{tabular}
\vspace{-0.1in}\vspace{-0.1in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: Long run relations are $\mathbf{{\Greekmath 010C} }_{j,0}^{\prime }\mathbf{w}
_{it}={\Greekmath 010C} _{j1,0}SD_{it}+{\Greekmath 010C} _{j2,0}LD_{it}+{\Greekmath 010C} _{j3,0}TA_{it}$, for $
j=1,2$, where $SD_{it}$, $LD_{it}$ and $TA_{it}$ are logarithms of
short-term and long-term debts, and total assets. The first long-run
relation $(j=1)$ is identified using the exact identifying conditions ${\Greekmath 010C}
_{11,0}=1$ and ${\Greekmath 010C} _{12,0}=0$. The second long-run relation ($j=2$) is
identified using the exact identifying conditions ${\Greekmath 010C} _{22,0}=1$ and $
{\Greekmath 010C} _{21,0}=0$. This table reports estimates for ${\Greekmath 010C} _{13,0}$ and $
{\Greekmath 010C} _{23,0}$ using PME estimator given by (\ref{BETAdot_hat}) with $q=2$
sub-sample time averages, subject to $T_{i}\geq 20$, for all $i$. To
simplify the notations we use tilde symbol to denote the exactly identified
PME estimates. See Section \ref{Sup_micro} of the supplement for variable
definitions, data sources and availability, and filters applied.\vspace{
-0.3in}
\end{flushleft}

\begin{center}
\singlespacing
TABLE 10:\textit{\ }PME estimation results for the variable set $\left\{
DO_{it},BV_{it},MV_{it}\right\} $, using $q=2$\ sub-sample time averages and
two\ long-run relations

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rcccccc}
\hline\hline
& \multicolumn{2}{c}{First exactly identified} &  &  & \multicolumn{2}{c}{
Second exactly identified} \\
& \multicolumn{2}{c}{long-run relation} &  &  & \multicolumn{2}{c}{long-run
relation} \\
\multicolumn{1}{l}{} & \multicolumn{2}{c}{$\mathbf{{\Greekmath 010C} }_{1,0}^{\prime }
\mathbf{w}_{it}=DO+{\Greekmath 010C} _{13,0}BV$} &  &  & \multicolumn{2}{c}{$\mathbf{
{\Greekmath 010C} }_{2,0}^{\prime }\mathbf{w}_{it}=BV+{\Greekmath 010C} _{23,0}MV$} \\ \hline
Sample end year: & 2021 & 2010 &  & Sample end year: & 2021 & 2010 \\ \hline
\multicolumn{1}{l}{$\tilde{{\Greekmath 010C}}_{13}$} & -1.055 & -0.990 &  &
\multicolumn{1}{l}{$\tilde{{\Greekmath 010C}}_{23}$} & -0.937 & -0.927 \\
\multicolumn{1}{l}{s.e.} & (0.019) & (0.020) &  & \multicolumn{1}{l}{s.e.} &
(0.008) & (0.009) \\
\multicolumn{1}{l}{t(${\Greekmath 010C} _{13,0}=-1)$} & 2.8 & -0.5 &  &
\multicolumn{1}{l}{t(${\Greekmath 010C} _{23,0}=-1)$} & -7.7 & -8.6 \\ \hline
\multicolumn{1}{l}{$n$} & 1,415 & 1,164 &  & $n$ & 1,415 & 1,164 \\
\multicolumn{1}{l}{$\sum_{i=1}^{n}T_{i}$} & 44,592 & 35,293 &  & $
\sum_{i=1}^{n}T_{i}$ & 44,592 & 35,293 \\ \hline\hline
\end{tabular}
\vspace{-0.1in}\vspace{-0.1in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: Long run relations are $\mathbf{{\Greekmath 010C} }_{j,0}^{\prime }\mathbf{w}
_{it}={\Greekmath 010C} _{j1,0}DO_{it}+{\Greekmath 010C} _{j2,0}BV_{it}+{\Greekmath 010C} _{j3,0}MV_{it}$, for $
j=1,2$, where $DO_{it}$, $BV_{it}$ and $MV_{it}$ are logarithms of total
debt outstanding, book, and market values. The first long-run relation ($j=1$
) is identified using the exact identifying conditions ${\Greekmath 010C} _{11,0}=1$ and
${\Greekmath 010C} _{12,0}=0$. The second long-run relation ($j=2$) is identified using
the exact identifying conditions ${\Greekmath 010C} _{22,0}=1$ and ${\Greekmath 010C} _{21,0}=0$.
See also the notes to Table 9.\pagebreak
\end{flushleft}

\normalsize
\onehalfspacing

Table 10 reports similar results for $\mathbf{w}_{it}=(DO_{it},$ $BV_{it},$ $
MV_{it})^{\prime }$ and two exactly identified long-run relations associated
with logarithms total debt, market value, and book value. PME estimates are
close to -1, but the unit long-run elasticity between pairs of variables in
this set is rejected for all cases except the logarithms of total debt
outstanding and market value in the sample ending in 2010.\footnote{
Rotation of the two identified long-run relationships reported in Table 10
imply the long-run relations $DO_{it}-0.888BV_{it}$ and $
DO_{it}-0.937BV_{it} $ for the samples ending in 2021 and 2010,
respectively, both of these PME estimates are statistically significanlty
different from -1 at the 1 percent level.}

Overall, the results support the existence of long-run relations between the
logarithms of a number of key financial variables considered in the
corporate finance literature. But with the notable exception of the
logarithm of short term debt to total asset ratio, the use of other
financial ratios as stationary variables in financial analysis is not
supported by our empirical findings. Instead our study recommends using
estimated long-run relations such as $DO_{i,t-1}-1.14$ $TA_{i,t-1}$, $
LD_{i,t-1}-1.19$ $TA_{i,t-1}$, and $BV_{i,t-1}-0.927MV_{i,t-1}$, as error
correction terms in dynamic panel regressors that allow for short term
dynamics and other stationary variables. This allows the analysis of short
term dynamics of financial variables to be coherently embedded in a long-run
equilibrating framework.

\subsection{International macro applications using Penn World Table}

Our second empirical application considers cross country macroeconomic time
series data from the Penn World Table\footnote{
We use version 10.01 of PWT database, available at
https://www.rug.nl/ggdc/productivity/pwt/, see
\shortciteN{FeenstraInklaarTimmer2015}
. See also the supplement for a detailed description of data constructions.}
(PWT), where $n$ (the number of countries) is smaller and the average time
dimension is larger as compared to the corporate finance data. Given the
good small sample performance of the PME approach even for smaller values of
$n$ in our Monte Carlo experiments, we have confidence in using the PME
estimation also in the case of the macro application. We focus on four key
macro variables, namely real merchandise exports per capita ($ex_{it}$),
real merchandise imports per capita ($im_{it}$), real productivity per hour
worked ($prod_{it}$) and real wages per hour worked ($wage_{it}$). The
choice of these variables was motivated by two widely maintained hypotheses.
Firstly, real wages and productivity should balance for steady state growth
to be feasible. Secondly export and imports should balance for international
solvency, though the constraint may not be binding for reserve-currency
countries such as the US. We first consider the two pairs that we expect to
cointegrate separately, then we consider all four variables together. As
with the corporate finance application, we are interested both in whether
they cointegrate and whether there is a unit coefficient. In both cases,
labour market balance and trade balance, there is no clear causal ordering
between the variables. Since the constraints that produce balance may be
somewhat different for advanced and emerging economies, we report estimates
for the country groupings separately as well as together.

The PME results for the analysis of a possible long-run relation between $
ex_{it}$ and $im_{it}$ are summarized in Table 11. This table gives the two
eigenvalues of $\mathbf{R}_{_{\bar{w}\bar{w}}}$ for $\mathbf{w}_{it}=\left(
ex_{it},im_{it}\right) ^{\prime }$ and $q=2$. (See (\ref{Rww})) As can be
seen there is a clear separation between the first and second eigenvalues
supporting the existence of a long-run relation between $ex_{it}$ and $
im_{it}$. This result holds for both choices of the threshold parameter $
{\Greekmath 010E} =1/2$ and $1/4$, and for all three country groupings. Given this
result we then estimated the long-run relation $\mathbf{{\Greekmath 010C} }^{\prime }
\mathbf{w}_{it}=$ ${\Greekmath 010C} _{11}ex_{it}+{\Greekmath 010C} _{12}im_{it}$, normalizing on $
{\Greekmath 010C} _{12}=1$, for all three country groupings. There is a marked
difference between the estimates of ${\Greekmath 010C} _{11}$ across the advanced and
emerging economies. For the advanced economies we have $\hat{{\Greekmath 010C}}
_{11}=-0.914$ ($0.029$), which strongly rejects the null hypothesis of $-1$
and suggests systematic differences can persist between merchandise imports
and exports for advanced economies. In contrast the estimate for the
emerging economies, namely $\hat{{\Greekmath 010C}}_{11}=-0.992$ ( $0.045$), is not
significantly different from $-1$. For all countries pooled, $\hat{{\Greekmath 010C}}
_{11}=$ $-0.972$ ($0.034$) and does not reject the null of ${\Greekmath 010C} _{11}=-1$.
\footnote{
Normalizing on ${\Greekmath 010C} _{11}$ yields the same results for ${\Greekmath 010C} _{12}$ since
PME estimate of ${\Greekmath 010C} _{12}$ is exactly equal to $1/\hat{{\Greekmath 010C}}_{11}$ and
by construction $Var(\hat{{\Greekmath 010C}}_{12})=Var(1/\hat{{\Greekmath 010C}}_{11})$.} The
difference in the estimates obtained for advanced and emerging economies
could be due to greater ability of advanced economies in financing their
goods trade imbalances by increasing their export of services and having
easier access to international capital market.

Similar results are obtained when we consider the relation between wages and
labour productivity. For all country groupings and both choices of the
thresholds we find $\tilde{r}=1$. See Table 12. The PME estimates of the
long-run relation, $\mathbf{{\Greekmath 010C} }^{\prime }\mathbf{w}_{it}=$ ${\Greekmath 010C}
_{11}wage_{it}+prod_{it}$ for advanced and emerging economies are $\hat{{\Greekmath 010C}
}_{11}=-0.953$ ($0.013$) and $\hat{{\Greekmath 010C}}_{11}=-0.984$ ($0.043$),
respectively. Once again estimates of ${\Greekmath 010C} _{11}$ are quite close to $-1$,
but as in the case of imports and exports the null of ${\Greekmath 010C} _{11}=-1$ is
rejected for advanced economies but not for emerging economies\textbf{.}

\begin{center}
\singlespacing
TABLE 11:\textit{\ }PME estimates for $\mathbf{w}_{it}=\left(
ex_{it},im_{it}\right) ^{\prime }$, using $q=2$\ sub-sample time averages.

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rccr}
\hline\hline
Sample: & Advanced & Emerging & All economies \\ \hline
\multicolumn{4}{l}{\textbf{Eigenvalues }of $\mathbf{R}_{_{\bar{w}\bar{w}}}$
\textbf{\ given by (\ref{Rww}) }(in ascending order)} \\ \hline
$\tilde{{\Greekmath 0115}}_{1}$ & 0.026 & 0.102 & \multicolumn{1}{c}{0.084} \\
$\tilde{{\Greekmath 0115}}_{2}$ & 1.974 & 1.898 & \multicolumn{1}{c}{1.916} \\ \hline
\multicolumn{4}{l}{\textbf{Threshold }$\bar{T}^{-{\Greekmath 010E} }$} \\ \hline
${\Greekmath 010E} =1/2$ & 0.128 & 0.134 & \multicolumn{1}{c}{0.132} \\
${\Greekmath 010E} =1/4$ & 0.357 & 0.366 & \multicolumn{1}{c}{0.364} \\ \hline
\multicolumn{4}{l}{\textbf{Estimated number of long-run relations (}$\tilde{r
}$\textbf{)}} \\ \hline
$\tilde{r}$ (${\Greekmath 010E} =1/2$) & 1 & 1 & \multicolumn{1}{c}{1} \\
$\tilde{r}$ (${\Greekmath 010E} =1/4$) & 1 & 1 & \multicolumn{1}{c}{1} \\ \hline
\multicolumn{4}{l}{\textbf{Exactly identified long-run relations}} \\
\multicolumn{4}{c}{$\mathbf{{\Greekmath 010C} }^{\prime }\mathbf{w}_{it}=\left(
\begin{array}{cc}
{\Greekmath 010C} _{11} & 1
\end{array}
\right) \left(
\begin{array}{c}
ex_{it} \\
im_{it}
\end{array}
\right) ={\Greekmath 010C} _{11}ex_{it}+im_{it}$} \\ \hline
$\hat{{\Greekmath 010C}}_{11}$ & -0.914 & -0.992 & \multicolumn{1}{c}{-0.972} \\
& (0.029) & (0.045) & \multicolumn{1}{c}{(0.034)} \\ \hline
\multicolumn{4}{l}{\textbf{Sample dimensions}} \\ \hline
$n$ & 38 & 139 & \multicolumn{1}{c}{177} \\
$\bar{T}=n^{-1}\sum_{i=1}^{n}T_{i}$ & 61.3 & 56.1 & \multicolumn{1}{c}{57.2}
\\ \hline\hline
\end{tabular}
\vspace{-0.1in}\vspace{-0.1in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: Standard errors are reported in parentheses. See Section \ref
{Sup_macro} of the supplement for variable definitions, data sources and
availability, and filters applied.
\end{flushleft}

\begin{center}
\normalsize
\singlespacing
TABLE 12:\textit{\ }PME estimates for $\mathbf{w}_{it}=\left(
wage_{it},prod_{it}\right) ^{\prime }$, using $q=2$\ sub-sample time
averages.

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rrrr}
\hline\hline
Sample: & Advanced & Emerging & All economies \\ \hline
\multicolumn{4}{l}{\textbf{Eigenvalues }of $\mathbf{R}_{_{\bar{w}\bar{w}}}$
\textbf{\ given by (\ref{Rww}) }(in ascending order)} \\ \hline
$\tilde{{\Greekmath 0115}}_{1}$ & \multicolumn{1}{c}{0.004} & \multicolumn{1}{c}{0.042}
& \multicolumn{1}{c}{0.015} \\
$\tilde{{\Greekmath 0115}}_{2}$ & \multicolumn{1}{c}{1.996} & \multicolumn{1}{c}{1.958}
& \multicolumn{1}{c}{1.985} \\ \hline
\multicolumn{4}{l}{\textbf{Threshold }$\bar{T}^{-{\Greekmath 010E} }$} \\ \hline
${\Greekmath 010E} =1/2$ & \multicolumn{1}{c}{0.135} & \multicolumn{1}{c}{0.146} &
\multicolumn{1}{c}{0.139} \\
${\Greekmath 010E} =1/4$ & \multicolumn{1}{c}{0.367} & \multicolumn{1}{c}{0.382} &
\multicolumn{1}{c}{0.373} \\ \hline
\multicolumn{4}{l}{\textbf{Estimated number of long-run relations (}$\tilde{r
}$\textbf{)}} \\ \hline
$\tilde{r}$ (${\Greekmath 010E} =1/2$) & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{1}
& \multicolumn{1}{c}{1} \\
$\tilde{r}$ (${\Greekmath 010E} =1/4$) & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{1}
& \multicolumn{1}{c}{1} \\ \hline
\multicolumn{4}{l}{\textbf{Exactly identified long-run relations}} \\
\multicolumn{4}{c}{$\mathbf{{\Greekmath 010C} }^{\prime }\mathbf{w}_{it}=\left(
\begin{array}{cc}
{\Greekmath 010C} _{11} & 1
\end{array}
\right) \left(
\begin{array}{c}
prod_{it} \\
wage_{it}
\end{array}
\right) ={\Greekmath 010C} _{11}prod_{it}+wage_{it}$} \\ \hline
$\hat{{\Greekmath 010C}}_{11}$ & \multicolumn{1}{c}{-0.953} & \multicolumn{1}{c}{-0.984}
& \multicolumn{1}{c}{-0.962} \\
& \multicolumn{1}{c}{(0.013)} & \multicolumn{1}{c}{(0.043)} &
\multicolumn{1}{c}{(0.016)} \\ \hline
\multicolumn{4}{l}{\textbf{Sample dimensions}} \\ \hline
$n$ & \multicolumn{1}{c}{35} & \multicolumn{1}{c}{24} & \multicolumn{1}{c}{59
} \\
$\bar{T}=n^{-1}\sum_{i=1}^{n}T_{i}$ & \multicolumn{1}{c}{55.5} &
\multicolumn{1}{c}{47.4} & \multicolumn{1}{c}{52.2} \\ \hline\hline
\end{tabular}
\vspace{-0.1in}\vspace{-0.1in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: See notes to Table 11.\vspace{-0.15in}
\end{flushleft}

\normalsize
\onehalfspacing

To illustrate how our proposed methods perform in the case of multiple
long-run relations we now consider all the four variables together and set $
\mathbf{w}_{it}=\left( ex_{it},im_{it},prod_{it},wage_{it}\right) ^{\prime }$
. Given the above pair-wise results we would expect at least two long-run
relations amongst these four variables. But the four eigenvalues of $\mathbf{
R}_{_{\bar{w}\bar{w}}}$ reported in Table 13 clearly suggest that there are
three long-run relations among the four variables, irrespective whether we
use ${\Greekmath 010E} =1/4$ or $1/2$ as the threshold parameter. This result
highlights the advantage of considering the possibility of multiple long-run
relations and can help with discovery of hitherto unnoticed or overlooked
long relations. For the third long-run relation we consider a possible
long-run relation between exports and productivity. There is extensive
microeconomic evidence suggesting exporting firms tend to have higher
productivity. See, for example,
\citeN{BernardJensen2004}
. However, there is controversy about whether the relation arises because
high productivity firms export more or whether the competitive pressure of
exporting boosts productivity. See for example
\shortciteN{Aghion_etal2018}
.\textbf{\ }Also, while the firm level micro relation has been examined for
many countries, the country level macro relation has been less intensively
explored. Our application provides cross country evidence on the relation
between exports and productivity without making any assumption about the
direction of causality between these variables. Accordingly, we consider the
following exactly identified long-run relations assuming that $r_{0}=3$
amongst the four variables $\mathbf{w}_{it}$,\vspace{-0.15in}
\begin{equation}
\mathbf{B}^{\prime }\mathbf{w}_{it}=\left(
\begin{array}{cccc}
{\Greekmath 010C} _{11} & 1 & 0 & 0 \\
0 & 0 & {\Greekmath 010C} _{23} & 1 \\
{\Greekmath 010C} _{31} & 0 & 1 & 0
\end{array}
\right) \left(
\begin{array}{c}
ex_{it} \\
im_{it} \\
prod_{it} \\
wage_{it}
\end{array}
\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}  \label{id4}
\end{equation}
PME estimates for ${\Greekmath 010C} _{11}$ and ${\Greekmath 010C} _{23}$ in Table 13 are in line
with the estimates in Tables 11 and 12. For the third long-run relation, we
obtain $\hat{{\Greekmath 010C}}_{31}=-0.509$ ($0.023$) for advanced economies and $\hat{
{\Greekmath 010C}}_{31}=-0.426$ ( $0.032$) for emerging economies. Hence, we discovered
that exports and productivity are related with the expected sign. Unlike the
other two long-run relations, we do not have any \textit{a priori} reason to
believe the estimates of ${\Greekmath 010C} _{31}$ should be close to $-1$.

To corroborate the evidence in Table 13 regarding the third long-run
relation, Table 14 reports PME findings for $\mathbf{w}_{it}=\left(
prod_{it},ex_{it}\right) ^{\prime }$. There is a clear separation of
eigenvalues, indicating existence of a long-run relation, in line with
findings in Table 14 for the four-variable vector $\mathbf{w}_{it}$. In
addition, the estimated coefficients in Table 14 are very similar to the
corresponding coefficients reported in Table 13.\vspace{-0.2in}

\begin{center}
\singlespacing
TABLE 13:\textit{\ }PME estimates for the variable set $\left\{
wage_{it},prod_{it},im_{it},ex_{it}\right\} $, using $q=2$\ sub-sample time
averages.

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rrrr}
\hline\hline
Sample: & Advanced & Emerging & All economies \\ \hline
\multicolumn{4}{l}{\textbf{Eigenvalues }of $\mathbf{R}_{_{\bar{w}\bar{w}}}$
\textbf{\ given by (\ref{Rww}) }(in ascending order)} \\ \hline
$\tilde{{\Greekmath 0115}}_{1}$ & \multicolumn{1}{c}{0.003} & \multicolumn{1}{c}{0.012}
& \multicolumn{1}{c}{0.014} \\
$\tilde{{\Greekmath 0115}}_{2}$ & \multicolumn{1}{c}{0.010} & \multicolumn{1}{c}{0.037}
& \multicolumn{1}{c}{0.015} \\
$\tilde{{\Greekmath 0115}}_{3}$ & \multicolumn{1}{c}{0.064} & \multicolumn{1}{c}{0.111}
& \multicolumn{1}{c}{0.088} \\
$\tilde{{\Greekmath 0115}}_{4}$ & \multicolumn{1}{c}{3.923} & \multicolumn{1}{c}{3.840}
& \multicolumn{1}{c}{3.883} \\ \hline
\multicolumn{4}{l}{\textbf{Threshold }$\bar{T}^{-{\Greekmath 010E} }$} \\ \hline
${\Greekmath 010E} =1/2$ & \multicolumn{1}{c}{0.135} & \multicolumn{1}{c}{0.146} &
\multicolumn{1}{c}{0.139} \\
${\Greekmath 010E} =1/4$ & \multicolumn{1}{c}{0.367} & \multicolumn{1}{c}{0.382} &
\multicolumn{1}{c}{0.373} \\ \hline
\multicolumn{4}{l}{\textbf{Estimated number of long-run relations (}$\tilde{r
}$\textbf{)}} \\ \hline
$\tilde{r}$ (${\Greekmath 010E} =1/2$) & \multicolumn{1}{c}{3} & \multicolumn{1}{c}{3}
& \multicolumn{1}{c}{3} \\
$\tilde{r}$ (${\Greekmath 010E} =1/4$) & \multicolumn{1}{c}{3} & \multicolumn{1}{c}{3}
& \multicolumn{1}{c}{3} \\ \hline
\multicolumn{4}{l}{\textbf{Exactly identified long-run relations}} \\
\multicolumn{4}{c}{$\mathbf{B}^{\prime }\mathbf{w}_{it}=\left(
\begin{array}{cccc}
{\Greekmath 010C} _{11} & 1 & 0 & 0 \\
0 & 0 & {\Greekmath 010C} _{23} & 1 \\
{\Greekmath 010C} _{31} & 0 & 1 & 0
\end{array}
\right) \left(
\begin{array}{c}
ex_{it} \\
im_{it} \\
prod_{it} \\
wage_{it}
\end{array}
\right) $} \\ \hline
$\hat{{\Greekmath 010C}}_{11}$ & \multicolumn{1}{c}{-0.882} & \multicolumn{1}{c}{-1.005}
& \multicolumn{1}{c}{-0.928} \\
& \multicolumn{1}{c}{(0.027)} & \multicolumn{1}{c}{(0.031)} &
\multicolumn{1}{c}{(0.023)} \\
$\hat{{\Greekmath 010C}}_{23}$ & \multicolumn{1}{c}{-0.952} & \multicolumn{1}{c}{-0.954}
& \multicolumn{1}{c}{-0.953} \\
& \multicolumn{1}{c}{(0.013)} & \multicolumn{1}{c}{(0.041)} &
\multicolumn{1}{c}{(0.015)} \\
$\hat{{\Greekmath 010C}}_{31}$ & \multicolumn{1}{c}{-0.509} & \multicolumn{1}{c}{-0.426}
& \multicolumn{1}{c}{-0.478} \\
& \multicolumn{1}{c}{(0.023)} & \multicolumn{1}{c}{(0.032)} &
\multicolumn{1}{c}{(0.021)} \\ \hline
\multicolumn{4}{l}{\textbf{Sample dimensions}} \\ \hline
$n$ & \multicolumn{1}{c}{35} & \multicolumn{1}{c}{24} & \multicolumn{1}{c}{59
} \\
$\bar{T}=n^{-1}\sum_{i=1}^{n}T_{i}$ & \multicolumn{1}{c}{55.5} &
\multicolumn{1}{c}{47.4} & \multicolumn{1}{c}{52.2} \\ \hline\hline
\end{tabular}
\vspace{-0.1in}\vspace{-0.1in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: See Section \ref{Sup_macro} of the supplement for variable
definitions, data sources and availability, and filters applied.\vspace{
-0.15in}
\end{flushleft}

\begin{center}
\normalsize
\singlespacing
TABLE 14:\textit{\ }PME estimates for $\mathbf{w}_{it}=\left(
ex_{it},prod_{it}\right) ^{\prime }$, using $q=2$\ sub-sample time averages.

\setlength{\tabcolsep}{5pt}
\scriptsize
\begin{tabular}{rrrr}
\hline\hline
Sample: & Advanced & Emerging & All economies \\ \hline
\multicolumn{4}{l}{\textbf{Eigenvalues }of $\mathbf{R}_{_{\bar{w}\bar{w}}}$
\textbf{\ given by (\ref{Rww}) }(in ascending order)} \\ \hline
$\tilde{{\Greekmath 0115}}_{1}$ & \multicolumn{1}{c}{0.025} & \multicolumn{1}{c}{0.080}
& \multicolumn{1}{c}{0.061} \\
$\tilde{{\Greekmath 0115}}2$ & \multicolumn{1}{c}{1.975} & \multicolumn{1}{c}{1.920} &
\multicolumn{1}{c}{1.939} \\ \hline
\multicolumn{4}{l}{\textbf{Threshold }$\bar{T}^{-{\Greekmath 010E} }$} \\ \hline
${\Greekmath 010E} =1/2$ & \multicolumn{1}{c}{0.135} & \multicolumn{1}{c}{0.146} &
\multicolumn{1}{c}{0.139} \\
${\Greekmath 010E} =1/4$ & \multicolumn{1}{c}{0.367} & \multicolumn{1}{c}{0.382} &
\multicolumn{1}{c}{0.373} \\ \hline
\multicolumn{4}{l}{\textbf{Estimated number of long-run relations (}$\tilde{r
}$\textbf{)}} \\ \hline
$\tilde{r}$ (${\Greekmath 010E} =1/2$) & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{1}
& \multicolumn{1}{c}{1} \\
$\tilde{r}$ (${\Greekmath 010E} =1/4$) & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{1}
& \multicolumn{1}{c}{1} \\ \hline
\multicolumn{4}{l}{\textbf{Exactly identified long-run relations}} \\
\multicolumn{4}{c}{$\mathbf{B}^{\prime }\mathbf{w}_{it}=\left(
\begin{array}{cc}
{\Greekmath 010C} _{11} & 1
\end{array}
\right) \left(
\begin{array}{c}
ex_{it} \\
prod_{it}
\end{array}
\right) ={\Greekmath 010C} _{11}ex_{it}+prod_{it}$} \\ \hline
$\hat{{\Greekmath 010C}}_{11}$ & \multicolumn{1}{c}{-0.510} & \multicolumn{1}{c}{-0.357}
& \multicolumn{1}{c}{-0.432} \\
& \multicolumn{1}{c}{(0.023)} & \multicolumn{1}{c}{(0.044)} &
\multicolumn{1}{c}{(0.036)} \\ \hline
\multicolumn{4}{l}{\textbf{Sample dimensions}} \\ \hline
$n$ & \multicolumn{1}{c}{35} & \multicolumn{1}{c}{29} & \multicolumn{1}{c}{64
} \\
$\bar{T}=n^{-1}\sum_{i=1}^{n}T_{i}$ & \multicolumn{1}{c}{55.5} &
\multicolumn{1}{c}{47.0} & \multicolumn{1}{c}{51.7} \\ \hline\hline
\end{tabular}
\vspace{-0.1in}\vspace{-0.1in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: See notes to Table 11.\vspace{-0.15in}
\end{flushleft}

\begin{landscape}

\begin{center}
\singlespacing
TABLE 15: Comparison of PME estimates with alternative estimates of single
long-run relation, $\mathbf{w}_{it}=\left( w_{1it},w_{2,t}\right) ^{\prime }$
\bigskip

\setlength{\tabcolsep}{2pt}
\scriptsize
\begin{tabular}{rcccrcccrcccrcccrcccrccc}
\hline\hline
$w_{1it}:$ & \multicolumn{7}{c}{$ex_{it}$} &  & \multicolumn{7}{c}{$
prod_{it} $} &  & \multicolumn{7}{c}{$ex_{it}$} \\
$w_{2,t}:$ & \multicolumn{7}{c}{$im_{it}$} &  & \multicolumn{7}{c}{$
wage_{it} $} &  & \multicolumn{7}{c}{$prod_{it}$} \\
\cline{2-8}\cline{10-16}\cline{18-24}
Coint. relation: & \multicolumn{3}{c}{${\Greekmath 010C} _{11}w_{1it}+w_{2,t}$} &  &
\multicolumn{3}{c}{$w_{1it}+{\Greekmath 010C} _{12}w_{2,t}$} &  & \multicolumn{3}{c}{$
{\Greekmath 010C} _{11}w_{1it}+w_{2,t}$} &  & \multicolumn{3}{c}{$w_{1it}+{\Greekmath 010C}
_{12}w_{2,t}$} &  & \multicolumn{3}{c}{${\Greekmath 010C} _{11}w_{1it}+w_{2,t}$} &  &
\multicolumn{3}{c}{$w_{1it}+{\Greekmath 010C} _{12}w_{2,t}$} \\
& \multicolumn{3}{c}{$\hat{{\Greekmath 010C}}_{11}$} &  & \multicolumn{3}{c}{$\hat{{\Greekmath 010C}}
_{12}$} &  & \multicolumn{3}{c}{$\hat{{\Greekmath 010C}}_{11}$} &  & \multicolumn{3}{c}{$
\hat{{\Greekmath 010C}}_{12}$} &  & \multicolumn{3}{c}{$\hat{{\Greekmath 010C}}_{11}$} &  &
\multicolumn{3}{c}{$\hat{{\Greekmath 010C}}_{12}$} \\
\cline{2-4}\cline{6-8}\cline{10-12}\cline{14-16}\cline{18-20}\cline{22-24}
Sample: & Adv. & Eme. & All & \multicolumn{1}{c}{} & Adv. & Eme. & All &
\multicolumn{1}{c}{} & Adv. & Eme. & All & \multicolumn{1}{c}{} & Adv. & Eme.
& All & \multicolumn{1}{c}{} & Adv. & Eme. & All & \multicolumn{1}{c}{} &
Adv. & Eme. & All \\ \hline
PME & -0.914 & -0.992 & -0.972 & \multicolumn{1}{c}{} & -1.094 & -1.008 &
-1.029 & \multicolumn{1}{c}{} & -0.953 & -0.984 & -0.962 &
\multicolumn{1}{c}{} & -1.049 & -1.016 & -1.039 & \multicolumn{1}{c}{} &
-0.510 & -0.357 & -0.432 & \multicolumn{1}{c}{} & -1.962 & -2.803 & -2.315
\\
& (0.029) & (0.045) & (0.034) & \multicolumn{1}{c}{} & (0.035) & (0.045) &
(0.036) & \multicolumn{1}{c}{} & (0.013) & (0.043) & (0.016) &
\multicolumn{1}{c}{} & (0.015) & (0.056) & (0.021) & \multicolumn{1}{c}{} &
(0.023) & (0.044) & (0.036) & \multicolumn{1}{c}{} & (0.075) & (0.251) &
(0.119) \\
SPMG & -0.962 & -1.026 & -0.976 & \multicolumn{1}{c}{} & -1.040 & -0.974 &
-1.025 & \multicolumn{1}{c}{} & -1.072 & -0.987 & -1.043 &
\multicolumn{1}{c}{} & -0.933 & -1.013 & -0.959 & \multicolumn{1}{c}{} &
-0.367 & -0.704 & -0.371 & \multicolumn{1}{c}{} & -2.729 & -1.421 & -2.697
\\
& (0.004) & (0.006) & (0.004) & \multicolumn{1}{c}{} & (0.004) & (0.006) &
(0.004) & \multicolumn{1}{c}{} & (0.005) & (0.004) & (0.003) &
\multicolumn{1}{c}{} & (0.005) & (0.004) & (0.003) & \multicolumn{1}{c}{} &
(0.004) & (0.013) & (0.003) & \multicolumn{1}{c}{} & (0.027) & (0.033) &
(0.024) \\
Breitung's 2-step & -0.842 & -0.924 & -0.902 & \multicolumn{1}{c}{} & -1.048
& -0.882 & -0.922 & \multicolumn{1}{c}{} & -1.144 & -1.010 & -1.066 &
\multicolumn{1}{c}{} & -0.832 & -0.924 & -0.900 & \multicolumn{1}{c}{} &
-0.522 & -0.371 & -0.455 & \multicolumn{1}{c}{} & -1.701 & -1.901 & -1.756
\\
& (0.010) & (0.021) & (0.016) & \multicolumn{1}{c}{} & (0.019) & (0.016) &
(0.013) & \multicolumn{1}{c}{} & (0.008) & (0.020) & (0.005) &
\multicolumn{1}{c}{} & (0.005) & (0.018) & (0.004) & \multicolumn{1}{c}{} &
(0.013) & (0.013) & (0.010) & \multicolumn{1}{c}{} & (0.042) & (0.095) &
(0.040) \\
PMG & -0.985 & -0.996 & -0.989 & \multicolumn{1}{c}{} & -0.973 & -0.949 &
-0.960 & \multicolumn{1}{c}{} & -1.140 & -0.990 & -1.100 &
\multicolumn{1}{c}{} & -0.837 & -0.962 & -0.886 & \multicolumn{1}{c}{} &
-0.291 & -0.718 & -0.306 & \multicolumn{1}{c}{} & -1.568 & -1.350 & -1.527
\\
& (0.007) & (0.008) & (0.005) & \multicolumn{1}{c}{} & (0.009) & (0.009) &
(0.006) & \multicolumn{1}{c}{} & (0.008) & (0.007) & (0.005) &
\multicolumn{1}{c}{} & (0.006) & (0.009) & (0.004) & \multicolumn{1}{c}{} &
(0.007) & (0.020) & (0.006) & \multicolumn{1}{c}{} & (0.026) & (0.077) &
(0.024) \\
PB & -0.868 & -0.813 & -0.828 & \multicolumn{1}{c}{} & -0.993 & -0.869 &
-0.891 & \multicolumn{1}{c}{} & -1.136 & -0.966 & -1.097 &
\multicolumn{1}{c}{} & -0.844 & -0.961 & -0.888 & \multicolumn{1}{c}{} &
-0.486 & -0.379 & -0.437 & \multicolumn{1}{c}{} & -1.688 & -1.967 & -1.771
\\
& (0.027) & (0.033) & (0.025) & \multicolumn{1}{c}{} & (0.039) & (0.037) &
(0.031) & \multicolumn{1}{c}{} & (0.006) & (0.035) & (0.002) &
\multicolumn{1}{c}{} & (0.005) & (0.031) & (0.002) & \multicolumn{1}{c}{} &
(0.028) & (0.056) & (0.038) & \multicolumn{1}{c}{} & (0.066) & (0.293) &
(0.103) \\
PFMOLS & -0.857 & -0.835 & -0.841 & \multicolumn{1}{c}{} & -1.083 & -0.886 &
-0.934 & \multicolumn{1}{c}{} & -1.241 & -0.978 & -1.070 &
\multicolumn{1}{c}{} & -0.776 & -0.944 & -0.896 & \multicolumn{1}{c}{} &
-0.528 & -0.329 & -0.445 & \multicolumn{1}{c}{} & -1.723 & -2.051 & -1.822
\\
& (0.023) & (0.035) & (0.026) & \multicolumn{1}{c}{} & (0.050) & (0.043) &
(0.032) & \multicolumn{1}{c}{} & (0.015) & (0.485) & (0.009) &
\multicolumn{1}{c}{} & (0.010) & (0.163) & (0.008) & \multicolumn{1}{c}{} &
(0.064) & (0.060) & (0.092) & \multicolumn{1}{c}{} & (0.059) & (0.191) &
(0.081) \\
PDOLS & -0.912 & -0.841 & -0.856 & \multicolumn{1}{c}{} & -1.045 & -0.801 &
-0.853 & \multicolumn{1}{c}{} & -1.093 & -1.029 & -1.089 &
\multicolumn{1}{c}{} & -0.868 & -1.029 & -0.891 & \multicolumn{1}{c}{} &
-0.520 & -0.409 & -0.470 & \multicolumn{1}{c}{} & -1.853 & -1.957 & -1.901
\\
& (0.004) & (0.006) & (0.004) & \multicolumn{1}{c}{} & (0.005) & (0.006) &
(0.004) & \multicolumn{1}{c}{} & (0.006) & (0.004) & (0.003) &
\multicolumn{1}{c}{} & (0.004) & (0.004) & (0.003) & \multicolumn{1}{c}{} &
(0.004) & (0.008) & (0.004) & \multicolumn{1}{c}{} & (0.015) & (0.038) &
(0.015) \\ \hline
\multicolumn{1}{l}{Sample size} &  &  &  & \multicolumn{1}{c}{} &  &  &  &
\multicolumn{1}{c}{} &  &  &  & \multicolumn{1}{c}{} &  &  &  &
\multicolumn{1}{c}{} &  &  &  & \multicolumn{1}{c}{} &  &  &  \\ \hline
$n$ & 38 & 139 & 177 & \multicolumn{1}{c}{} & 38 & 139 & 177 &
\multicolumn{1}{c}{} & 35 & 24 & 59 & \multicolumn{1}{c}{} & 35 & 24 & 59 &
\multicolumn{1}{c}{} & 35 & 29 & 64 & \multicolumn{1}{c}{} & 35 & 29 & 64 \\
$\bar{T}=n^{-1}\sum_{i=1}^{n}T_{i}$ & 61.3 & 56.1 & 57.2 &
\multicolumn{1}{c}{} & 61.3 & 56.1 & 57.2 & \multicolumn{1}{c}{} & 55.5 &
47.4 & 52.2 & \multicolumn{1}{c}{} & 55.5 & 47.4 & 52.2 & \multicolumn{1}{c}{
} & 55.5 & 47.0 & 51.7 & \multicolumn{1}{c}{} & 55.5 & 47.0 & 51.7 \\
\hline\hline
\end{tabular}
\vspace{-0.1in}\vspace{-0.1in}
\end{center}

\begin{flushleft}
\scriptsize
\singlespacing
Notes: PME and SPMG are the only estimators where $\hat{{\Greekmath 010C}}_{11}=\hat{
{\Greekmath 010C}}_{12}^{-1}$. This is not the case for the remaining estimators. PME is
the pooled minimum eigenvalue estimator using $q=2$ subsamples. SPMG\ is the
system PMG estimator of
\citeN{ChudikPesaranSmith2023}
, using $p=2$ lags in levels ($1$ lag in first differences). Breitung's
2-step is the two-step system estimator of
\citeN{Breitung2005}
, using $p=2$ lags in levels. PMG\ is the Pooled Mean Group (PMG) estimator
by
\citeN{PesaranShinSmith1999}
, using $p=2$ lags in levels. PB is the pooled Bewley estimator by
\citeN{ChudikPesaranSmith2021PB}
, using $p=2$ lags in levels. PFMOLS is the panel FMOLS estimator by
\citeANP{Pedroni1996} (\citeyearNP{Pedroni1996}, \citeyearNP{Pedroni2001}, \citeyearNP{Pedroni2001ReStat})
. PDOLS is the panel Dynamic OLS (PDOLS) by
\citeN{MarkSul2003}
, using one lead and one lag of first differences.
\end{flushleft}

\end{landscape}

\normalsize
\onehalfspacing

\subsection{Comparison of PME and their estimates in the case of pair-wise
relations}

Here we provide comparative estimates for the pair-wise long-run relations
for which a number of alternative estimators are proposed in the literature.
Specifically, we compare PME estimates reported in Tables 11, 12, and 14
with the SPMG, Breitung's 2-step, PMG, PB, FMOLS, and PDOLS estimators
discussed in Sub-Section \ref{SubS_MC_c}.\footnote{
The SPMG estimator is proposed by
\citeN{ChudikPesaranSmith2023}
, the Breitung's two-step system estimator is proposed by
\citeN{Breitung2005}
, the PMG\ estimator is proposed by
\citeN{PesaranShinSmith1999}
, the PB (Pooled Bewley) estimator is proposed by
\citeN{ChudikPesaranSmith2021PB}
, the panel FMOLS\ estimator is proposed by
\citeANP{Pedroni1996} (\citeyearNP{Pedroni1996}, \citeyearNP{Pedroni2001}, \citeyearNP{Pedroni2001ReStat})
, and the panel DOLS estimator is proposed by
\citeN{MarkSul2003}
.}\textbf{\ }The results are summarized in Table 15. PME and SPMG are the
only estimators that are invariant to the normalization imposed on $\mathbf{
{\Greekmath 010C} }^{\prime }\mathbf{w}_{it}={\Greekmath 010C} _{11}w_{1,it}+{\Greekmath 010C} _{12}w_{2,it}$.
As a result, the equality $\hat{{\Greekmath 010C}}_{11}\hat{{\Greekmath 010C}}_{12}=1$ holds only in
the case of PME and SPMG estimators.

The estimates of ${\Greekmath 010C} _{11}$ and ${\Greekmath 010C} _{12}$ are generally similar but
there are some large differences. For instance, the PME estimate of the
long-run coefficient in ${\Greekmath 010C} _{11}ex_{it}+prod_{it}$ in the case of
advanced economies is $-0.510$ ($0.023$) compared with SPMG estimate of $
-0.367$ ($0.004$). Such difference could arise from the possibility that
some of the SPMG modeling assumptions regarding short-run dynamics are not
met, whereas the PME estimator is more general in that it does not require
modeling of short-run dynamics. We also observe that standard errors of the
estimates proposed in the literature are in some cases 2 to 10-fold smaller
compared to PME. As we have seen in Monte Carlo experiments all of the
estimators proposed in the literature severely over-reject the null, whilst
this is not the case if we consider the MC results for size reported in
Table 6.

\section{Concluding remarks\label{CON}}

This paper provides a new, pooled minimum eigenvalue (PME), methodology for
the analysis of multiple long-run relations in panel data models where the
cross section dimension, $n$, is large relative to the time series
dimension, $T$. \ It uses non-overlapping sub-sample time averages as
deviations from their full-sample counterpart and estimates the number of
long-run relations and their coefficients using eigenvalues and eigenvectors
of the pooled covariance matrix of these sub-sample deviations. It applies
to unbalanced panels generated from general linear processes with
interactive stationary time effects and does not require knowing long-run
causal linkages. The PME estimator is consistent and asymptotically normally
distributed as $n$ and $T$ $\rightarrow \infty $ jointly, such that $
T\approx n^{d}$, with $d>0$ for consistency and $d>1/2$ for asymptotic
normality. Extensive Monte Carlo studies show that the number of long-run
relations can be estimated with high precision and the PME estimates of the
long-run coefficients show small bias and $RMSE$ and have good size and
power properties. The utility of our approach is illustrated with both micro
and macro applications. The micro application uncovers long-run relations
among key financial variables in an unbalanced panel of US firms from merged
CRSP-Compustat data set covering $2,000$ plus firms over the period $
1950-2021$. The macro application uses cross country macroeconomic time
series data covering up to $177$ countries over slightly shorter time
period, $1950-2019.$

As well as application in other areas than corporate finance and
macroeconomics the procedure opens up a range of theoretical developments.
One question that we are considering investigating is whether it is possible
to develop a unit root test based on similar principles. Our method is
semi-parametric, in that it does not model the short run dynamics. Because
the estimates of the long-run relations are super consistent, then having
estimated them, the PME estimates of the long-run relations could be used as
inputs into second stage models using stationary variables. For instance,
they can provide estimates of the disequilibrium terms in error correction
models of the short run dynamics, to measure speeds of adjustment. Such
models could also be used for medium-term forecasting and counterfactual
analysis.

\newpage

\noindent